,# LIBRARY OF COXGRKSS. t 



U # 

J UNITED STATES OF AMERICA.^ 



HYDRAULIC MOTORS 



r-f 



TRANSLATED FEOM THE 



FRENCH COtJRS DE MECANIQTJE APPLIQXJEE. 






M. "BR ESSE 

Professeur de Mecanique a V^^cole des Fonts et Chattssees. 



F. A. MAHA^, 

LIEUTENANT TJ. 8. C0KP8 OF ENGINEEE6. 



EEVISED BY 



D. H. MAHAN, LL.D., 

PE0FES80K OF CIVIL ENGINEERING, &C., UNITED STATES MILITARY ACADEMY. 




NEW YORK : 
JOHN WILEY & SON, 2 CLINTON HALL, ASTOR PLACE. 

1869. 



TT<< 



Entered according to Act of Congress, in the year 1869, by 

F, A. MAHAN, 

In the Clerk's Oflace of the District Court of the United States for the Southern District of 

New York. 



4 



f 



i'^^p^ 



y 



The New York Printing Company, 
8i, 83, and 85 Centre Street, 

New York. 



PREFACE 



The eminent position of M. Bresse in the scientific 
world, and in tlie French Corps of Civil Engineers, is 
my best apology for attempting to supply a want, felt 
by tlie students of civil engineering in our country, of 
some standard work on Hydraulic Motors, by fui-nisli- 
ing a translation of the chapter on this subject con- 
tained in the second volume of M. Bresse's lectures 
on Applied Mechanics, delivered to the pupils of the 
School of Civil Engineers {ilcole des Fonts et CJiaus- 
sees) at Paris. 

In making the translation, I have retained the French 
units of weights and measui^es in the numerical exam- 
ples given, as the majority of our students are conver- 
sant with them. 

F. A. MAHAN. 
Willett's Point, N". Y., July, 1869. 



CONTENTS. 



§ I.— PEELIMINARY IDEAS ON HYDRAULIC MOTORS. 

AKT. PAGE. 

Definitions ; theorem of the transmission of work in machines 1 9 

Analogous ideas applied to a waterfall 2 11 

General observations on the means of securing a good effective 

delivery for a waterfall driving a hydraulic motor 3 14 



§ II._WATER WHEELS WITH A HORIZONTAL AXLE. 

Undershot wheel with plane buckets or floats moving in a confined 

race 4 17 

Wheels arranged according to Poncelet's method 5 28 

Paddle wheels in an unconfined current 6 34 



Breast Wheels. 

Wheels set in a circular race, called breast wheels 7 86 

Example of calculations for a rapidly moving breast wheel 8 48 

Overshot Wheels. 

Wheels with buckets, or overshot wheels 9 51 

§ III.— WATER WHEELS WITH VERTICAL AXLES. 

Old-fashioned spoon or tub wheels 10 6Q 

Turbines. 

Of turbines 11 69 

Foumeyron's turbine 12 70 

Fontaine's turbine 13 73 

Koecklin's turbine 14 76 

Theory of the three preceding turbines 15 78 

Kemarks on the angles /?, y, 9, and on the dimensions b, b', r, r\ 

h, h' 16 90 



Vlll CONTENTS. 

ART. PAGE. 

Examples of the calculations to be made in constructing a turbine. .17 93 

Of the means of regulating the expenditure of water in turbines. . . 18 98 

Hydropneumatic turbine of Girard and Gallon 19 103 

Some practical views on the subject of turbines 20 105 

Reaction wheels 31 107 



§ IV.— OF A FEW MACHINES FOR RAISING WATER. 

Pumps 22 111 

Spiral noria 23 122 

Lifting turbines ; centrifugal pump 24 127 

Authorities on Water Wheels 134 



APPENDIX. 

Comparative table of French and United States measures 135 

Note A, Article 1 135 

Note B, Article 2 136 

Note C, Article 4 137 

Note D, Article 9 140 

Note E, Boyden turbines, from LoiDell Hydraulic Experiments^ by 

James B. Francis, Esq 141 






HYDRAULIC MOTORS, 



AND SOME 



MACHINES FOE EAISING WATER. 



§ I. — Pkeliminailt Ideas on Hydraulic Motors. 

1. Definitions / theorem of the transmission of work in ma- 
chines. — The term machine is applied to any body or collection 
of bodies intended to receive at some of their points certain 
forces, and to exert, at other points, forces which generally 
differ from the first in their intensity, direction, and the velocity 
of their points of application. 

The dynamic effect of a machine is the total work, generally 
negative or resisting, which it receives from external bodies 
subject to its action. It happens that the dynamic efi*ect is 
sometimes positive work: for example, when we let down a 
load by a rope passed over a pulley, the weight of the load pro- 
duces a motive work on the system of the rope and pulley. 

Let us suppose, to make this clear, that the dynamic effect is 
a resisting work. Independently of this, the machine is afiected 
by some others which are employed to overcome friction, the 
resistance of the air, &c. The resistances which give rise to 
these negative works have received the name of secondary 



10 GENERAL THEOREM. 

resistances I whilst the dynamic effect is due to what are called 
principal resistances. 

The genera] theorem of mechanics, in virtue of which a 
relation is established between the increase of living force of 
a material system and the work of the forces, is applicable 
to a machine as to every assemblage of bodies. To express 
it analytically, let us suppose the dynamic effect to be taken 
negative, and let us call 

T^, the sum of the work of the motive forces which have 
acted on the machine during a certain interval of time ; 

Tg, the dynamic effect during the same time ; 

Ty, the corresponding value of the work of the secondary 
resistances ; 

V and ^Jo, the velocities of a material point, whose mass is m, 
making a part of tlie machine, at the beginning and end 
of the time in question ; 

H and Ho, the corresponding distances of the centre of grav- 
ity of the mechanism below a horizontal plane ; 

2, a sum extended to all the masses m. 

From the theorem above mentioned there obtains 
i2m ^^ - 42m < = T^ - T,- T^+ (H — Ho)2m^. 

In a machine moving regularly, each of the velocities v in- 
creases from zero, the value corresponding to a state of rest, to 
a certain maximum which it never exceeds ; the first member 
of the equation has, then, necessarily, a superior limit, below 
which it will be found, or which it will, at the farthest, reach, 
whatever be the interval of time to which the initial and final 
values Vq and v are referred. The same holds with the term 
(H — Ho) ^ m g^ when the machine moves without changing 
its place, as its centre periodically occupies the same positions. 
On the contrary, the terms T^, T^, T^ increase indefinitely with 
the time, if the motion of the machine is prolonged, because 



GENEKAL THEOEEM. 11 

new quantities of work are being continually added to those 
already accumulated. These terms will at length greatly 
surpass the others, so the equation, therefore, should, after an 
unlimited interval of time, reduce to 

T^ = Tg -f- Ty-. 
This is what would really take place, without supposing the 
time unlimited, if the beginning and end of this interval cor- 
responded to a state of rest of the machine, and if, at the same 
time, H = H^. We can then say that, as a general rule, the 
motive work is equal to the resisting work ; but as this last 
includes, besides the dynamic effect for which the machine is 
established, the work of the secondary resistances, we see that 
the action of the motive power is not all usefully employed, 
since a portion goes to overcoming the work of T^. 

T 

It is evident that the ratio ^ measures the proportional 

T T 

loss ; the ratio —-, or 1 — tj^, gives, on the contrary, a clear 

-*- m ^ ill 

idea of the portion of the effective work. This last ratio is 
what is called the delivery of the machine; it is evidently 
always less than unity, since in the best-arranged machines Ty 
still preserves a certain value. The skill of the constructor is 
shown in brmging this as near unity as possible. 

2. Analogous ideas applied to a vxiterfall. — A waterfall 
may be considered as a cmTent of water flowing through two 
sections C B, E F (Fig. 1), at no great distance apart, but 
with a noticeable difference of level, H, between the surface 
slope at C and E, and which may be assumed to yield a 
constant volume of water for each unit of time. The mate- 
rial liquid system thus comprised between the sections C B, 
E F, at each instant may be regarded as a machine that is con- 
tinually renewed, the molecules which flow out through E F 



12 



GENEEAL THEOEEM. 



being replaced by those which enter through C B. The motive 
work in this case will be that of the weight combined with that 
of the pressures on the external boundaries of the system ; the 
dynamic effect will be the work of the resistances against the 
fall of water caused by any apparatus whatever, a water-wheel 




Fig. 1. 

for example, exposed to its action. In its turn the water-wheel, 
considered as a machine, will receive from the fall a motive 
work sensibly equal to the dynamic effect that we have just 
spoken of,*^ and will change only a portion of it into useful 
work, which will be its dynamic effect proper. But in what is 
to follow we shall limit ourselves to studying the dynamic effect 
of the fall, and not that of the wheel. 

Although the motion of the liquid cannot always be strictly 
the same, because the wheel does not always maintain exactly 
the same position, still it can be so regarded without material 
error ; for after an interval of time ^, generally very short, 
occupied by a float or paddle in taking the place of the one that 
preceded it, everything returns to the same condition as at the 
beginning of the interval. Supposing the motion of the wheel 
regular and the paddles to be uniformly distributed, there is 
such a frequent periodicity in the state of the system that it 

* We say sensibly^ for the equality between the mutual actions of the watei 
and wheel do not involve that of corresponding- work. This equality is only 
strict in supposing the friction of the liquid against the solid sides of the 
wheel zero, which friction is in reality very slight. 



GENEKAL THEOKEM. 13 

almost amounts to a permanency. We will now apply Ber- 
nouilli's theorem to any molecule whatever, having a mass m, 
which, departing from the section C B with the velocity Yg, 
reaches E F with a velocity Y. The entire head is H, if we 
allow the parallelism of the threads in the extreme sections, for 
the pressure then varies according to the hydrostatic law in each 
of the two surfaces C B and E F, so that the points C and E 
can be considered as piezometric levels for the initial and final 
positions of the mass m. Let —t^ and —t/he the respective 
work referred to the unit of mass, and considered as resistances 
which m has encountered in its course between C B and E F, 
in consequence of the action of the wheel and of viscosity. 
Then from the general theorem of living forces we have the 
equation — 

1 Y^ — Yn' 

whence, 

/ Yq" Y'\ 

Consequently we see that the weight ?7i g of each molecule 
which passes from C B to E F gives rise to a dynamic effect 
m t^^ the value of which is expressed in the second member of 
the equation. The sum of the weights of the molecules m in- 
cluded in the entire weight P, which the current expends in a 
second, will produce a dynamic effect equal to a sum 2 of analo- 
gous expressions extended to all these masses ; considering Yq 
and Y as constant velocities in the respective sections C B and 
E F, this summation will give — 

Moreover, in each second a new weight P is supplied by the 
current ; there is then produced a new dynamic effect 2 m ^g, 
which thus represents the mean dynamic effect in each second. 



14: GENEEAL THEOREM. 

The quantity P (^ H + ^ — — ) reduces to P H, in the 

case in which Yq and Y are sensibly equal to zero, which hap- 
pens in measuring the difference of level between the basins 
from which the water starts and that into which it flows, when 
the water is nearly at a stand-still ; we then call the product 

P H the effective delivery of the fall. The ratio ~p~^ is the 

productive force • ^ m tf is the work lost. Dividing the last 
equation by P or 2 m g^ and supposing Y = 0, Yq = 0, there 
obtains, 

H in this expression may be regarded as the total head of water; 
— - — « the prodicctive head, that is, the height which, multi- 
plied by the weight P expended, would give the dynamic effect 

^ 7n tf 

per second : the head lost, to be subtracted from the 

^ ^ g ^ 771 ' 

total head when the productive head is required. We see that 
— - — ^ is the mean loss of head experienced by the molecules in 

g ^ 771 r J 

their passage from C B to E F ; for this expression represents 
exactly the mean work of viscosity on a molecule, referred to 
the unit of mass and divided by g. 

3. General remarks on the 7neans of securing a satisfactory 
delivery froTTi a head of water Tjohich moves an hydraulic 
motor. — In order to obtain a good satisfactory delivery, we 
must seek to diminish as much as possible the term 2 m tf, or 

2 771 t 

the mean loss of head •^- A few of the causes that pro- 

g ^ m ^ 

duce this loss will now be pointed out, and the manner in 
which they may be diminished. 



GENERAL THEOEEM. 15 

"Firstly, if the water enters the wheel, and can in consequence 
act on it, it is because it possesses a certain relative velocity w ; 
now, in the majority of cases this relative velocity gives rise to 
a violent agitation of the liquid and to vibratory motions, from 

which a loss of head is experienced equal to ^, like to what 

has been observed to obtain in the collision of solid bodies ; 

hence ^r- would be a portion of the head lost.* It is, then, 

generally a matter of importance to make the water enter with 
as small a relative velocity as possible. However, that is not 
necessary when w has its direction tangent to the sides that it 
comes in contact with, and when the particular arrangement 
of the apparatus allows the water to continue its relative mo- 
tion in the wheel without there being any shock of the threads 
on the solid sides or on the liquid already introduced, since 
then we have no longer to fear the violent disturbance that we 
have just spoken of. 

When the water on leaving the wheel is received into a race 
of invariable level, in which it loses its absolute velocity of 

* Let us suppose tliat the water tliat has once entered the wheel passes at 
once to relative rest : the destruction of the velocity w being then attributa- 
ble only to the resisting work of the molecular actions, this work for a fluid 
molecule having the mass m would be ^ m ijf^ a quantity which, when referred 

to the unit of mass and divided by g, would give the loss of head -- , which 

is the same for all the molecules. But the supposition of the instant produc- 
tion of relative rest is not, strictly speaking, true; weight, for example, can 
combine sometimes with molecular actions to bring about this result at the 

end of a sensible time. Consequently the expression r— must be considered 

^9 

less as the exact value of the quantity whose value we here wish to find, than 
as a superior limit which we approach more or less nearly, according to the 
particular case considered. 



16 GENERAL THEOREM. 

departure v\ we readily see that this velocity v' must be re- 
duced as much as possible. In fact, the water that leaves the 

wheel carries with it, in each second, a living force. 



2^ 
which might have been taken up by a resisting work of the 

hydraulic motor, and have thus increased by its amount the 
dynamic effect Tg ; whereas this living force will only serve to 
produce a disturbance and eddies in the interior race, and will 
enter the term 2 m tj. There are cases, however, in which we 
are obliged to give -y' a value more or less considerable ; we 
shall see presently, by a few examples, how it is sometimes 
possible to diminish this unfavorable condition. 

Ordinarily, the considerations above mentioned are under- 
stood when we say that the water must enter without shock, 
and leave without velocity. We can also add, that it is well 
not to deliver it too rapidly through channels of too little 
breadth, as this would involve losses of head to be included in 

2 m tf 

the expression -. 

^ g ^ 7)% 

We shall now proceed to examine the most widely known 
hydraulic motors, keeping principally in view the best means 
of making use of the head in each case, and showing the man- 
ner of calculating the dynamic effect that can be realized with 
the means adopted. 

Water-wheels are divided into two great classes — those hav- 
ing a vertical and those having a horizontal axis. The varieties 
included in these two classes will form the subject of the fol- 
lowing paragraphs. 



UNDERSHOT WHEELS. 



||. Water- Wheels with a Horizontal Axis. 



4. Undershot wheel with plane tuckets or floats moving in a 
confl/nedrace. — These wheels are ordinarily constructed of wood. 
Upon a polygonal arbor A (Fig. 2) a socket C, of cast iron, 




Fig. 2. 



is fastened by means of wooden wedges 5. Arms D are set 
in grooves cast in the socket, and are fastened to it by bolts ; 
these arms serve to support a ring E E, the segments of which 
are fastened to each other and to the arms by iron bands. In the 
ring are set the projecting pieces FF. . . ., of wood, placed at 



18 UNDEESHOT WHEELS. 

equal distances apart, and intended to support the floats G G , 

which are boards varying from 0'^.02 to 0°^.03 in thickness, 
situated in planes passing through the axis of the wheel and 
occupying its entire breadth. A single set of the foregoing 
parts would not be sufficient to give a good support to the 
floats. In wheels of little breadth in the direction of the axis 
two parallel sets will suffice ; if the wheels are broad, three or 
more may be requisite. 

The number of arms increases with the diameter of the 
wheel. In the more ordinary kinds, of 3 to 5 metres in dia- 
meter, each socket carries six arms. The floats may be about 
O'^.SS to O'^AO apart, and have a little greater depth in the 
direction of the radius, say 0^.60 to 0°^.T0. 

From this brief description of the wheel, let us now see how 
we can calculate the work which it receives from the head of 
water. The water flows in a very nearly horizontal current 
through a race B G H F (Fig. 3), of nearly the same breadth 




as the wheel, a portion G H of the bottom being hollowed 
out, in a direction perpendicular to the axis, to a cylindrical 
shape, and allowing but a slight play to the floats. The liquid 
molecules have, when passing C B, a velocity v, but shortly 
after they are confined in the intervals limited by two consecu- 
tive floats and the race. They entered these spaces with a 
mean relative velocity equal to the difference between the hori- 
zontal velocity v, and the velocity v' of the middle of the im- 
mersed portion of the floats, the direction of which last velocity 
is also very nearly horizontal. There result from this relative 



UNDEESHOT WHEELS. 19 

velocity a shock and disturbance which gradually subside, 
while the floats are traversing over the circular portion of the 
canal ; so that if this circular portion is sufficiently long, and 
if there be not too much play between the floats and the canal, 
the water that leaves the wheel will have a velocity sensibly 
equal to v\ The action brought to bear by the wheel on the 
water is the cause of the change in this velocity from v to v\ 
which gives us the means, as we shall presently see, of calculat- 
ing the total intensity of this action. 

For this purpose let us apply to the liquid system included 
between the cross sections C B, E F, in which the threads are 
supposed parallel, the theorem of quantities of motion projected 
on a horizontal axis. Kepresent by 

h the constant breadth of the wheel and canal : 



h, h! the depths C B, E F, of the extreme sections which are 

supposed to be rectangular ; 
F the total force exerted by the wheel on the water, or in- 
versely, in a horizontal direction ; 
P the expenditure of the current, expressed in pounds, per 

second ; 
n the weight of a cubic metre of the water ; 
^ the short interval of time during which C B E F passes 

to C' B^ E^ F^ 
The liquid system C B E F, here under consideration, is 
analogous to the one treated in Note A (see Appendix), in 
which a change in the surface level takes place ; and the man- 
ner of determining the gain in the quantity of motion during 
a short time ^, and calculating the corresponding impulses, 
during the same time, produced by gravity and the pressures 
on the exterior surface of the liquid system, are alike in both 
cases. 

Employing the foregoing notation, we obtain 
2 



20 UNDERSHOT WHEELS. 

1st. For the mean gain in the projected quantity of motion, 

— {v^-v); 

2d. For the impulses of the weight and pressures together, 
also taken in horizontal projection, ^U h & (h^ — h'^) ; to these 
impulses is to be added that produced by F, or — F ^, to have 
the sum of the projected impulses. 

We will then have 

— (^^-v) = in5a(A^-A^»)-Fd, 

whence we obtain 

¥ = ^{v-v')-onl) ^ {h"-h'). 

The forces of which F is the resultant in horizontal projec- 
tion are exerted in a contrary direction by the water on the 
wheel, at points whose vertical motion is nearly null, and 
whose horizontal velocity is approximately v' . The wheel will 
then receive from these forces, in each unit of time, a work 
F v\ which represents the dynamic effect T^ to within a slight 
error. So that 

T, = Fv' = -^'('y - v') - 1 n J v'(A'' - K'\ 

Moreover we have 

whence 

or finally, observing that t? = — 

T, = ?.'(.-.')-*PA&-j')....(l) 



UNDEESHOT WHEELS. 21 

In order that this formula should be tolerably exact, the depths 
A and h! must be quite small, without which the floats would 
make an appreciable angle with the vertical at the moment 
they leave the water ; the velocity of the points at which are 
applied the forces, whose resultant is F, could no longer be con- 
sidered horizontal, as heretofore, and a resistance due to the 
emersion of the floats would be produced, on account of the 
liquid uselessly raised by them. The water must also be con- 
fined a sufiiciently long time to assume the velocity v' . 

We can consider in formula (1), v and A as fixed data, and 
seek the most suitable value of the velocity v' of the floats, to 

make T^ a maximum. For this purpose, if we place — = a?, 
Tg = A P ^ — , formula (1) becomes 



A = 2.(l-.)-|,^(l-4 



and we must choose a? so as to make A a maximum. This 
value will be obtained by taking the value which reduces the 

d A 

first differential co-efficient -i — to zero, which gives the equa- 
tion 



lue of a 
mately 



The value of x deduced depends on -^-\ we find approxi- 



^^^ = 0.00 


X = 0.500 


A = 


= 0.500 


0.05 


0.553 




0.431 


0.10 


0.595 




0.373 



Generally we neglect the term ^-^ ( xj; then we have 



22 UNDEKSHOT WHEELS. 

X = 0.50, A = 0.50 ; and recollecting that the maximum value 

of A is very much chano-ed, even for small values of -^. Ee- 

sides, experiment does not show that there is any use in taking 
X greater than 0.50, as we have just found ; it would ratlier 
show a ratio of v' to v of about 0.4, which obtains probably 
because of the motion of the wheel being too swift to allow the 
liquid to pass completely from the velocity v to v\ during the 
time that it remains between the floats : a portion of the water 
passes without producing its entire dynamic effect, and the 
formula (1) ceases to conform to fact. 

Smeaton, an English engineer, made some experiments, in 
1759, on a small wheel, 0°^.609 in diameter, having plane floats. 
This wheel was enclosed in a race having a flat bottom, which 
is a defect, because the intervals included between the floats 
and the race are never completely closed. The weight P varied 
from 0^86 to 2.84 kilogrammes. In each experiment the work 
transmitted to the wheel was determined by the raising of a 
weight attached to a rope which was wound around the axle. 

v' 
The most suitable value oi' x = — was thus found to vary be- 
tween 0.34 and 0.52, the mean being 0.43. The number A, 
comprised betweea 0.29 and 0.35,hadfor a mean value about -J. 

We can then take definitively the ratio — = 0.4, as found by 

V 

experiment. The expression for A becomes then 
A = 0.48 - 2.1 ^—, 

which, for -^ = 0.05 and^ = 0.10, gives the numbers 

A = 0.375 and A = 0.27, 
very nearly those found by Smeaton. 



UNDERSHOT WHEELS. 23 

A few remarks remain to be made, to which it would be well 
to pay attention in practice. 

1st. It is well, as far as possible, to have the depth A say 
from 0°^.15 to 0°'.20. Too small a thickness of the stratum of 
water which impinges on the wheel would give a relatively ap- 
preciable importance to the unavoidable pla-y between the 
wheel and the race, a play which results in pure loss of the 
motive water. Too great a thickness has also its inconve- 

niences : for from the relation ^ = ^, it follows that for —. = 

h ^ ' v' 

0.4 and h = 0°^.20, we will then have 

065 _ 
^ - 0.04 - ^ -^^ ' 

the floats will then be immersed 0°^.50, adopting the thickness 
of 0°^.20, and were they more so, they would meet with consi- 
derable resistance in leaving the water, as we have already said. 
It is necessary then that h be neither too large nor too small : 
the limits of 0°^.15 to 0°'.20 are recommended by M. Belan- 
ger. 

2d. We should avoid as much as possible losses of head dur- 
ing the passage of the water from the basin up-stream to its 
arrival at the section C B near the wheel, losses which result 
in a diminution of effective delivery (No. 2). In order to give 
it the shape of a thin layer, from 0"^.15 to 0°^.20 in thickness, 
the water is made to flow under a sluice through a rectangular 
orifice : care should be taken to avoid abrupt changes of direc- 
tion between the sides of this orifice and the interior of the 
dam, in order to avoid a contraction followed by a sudden 
change of direction of the threads, as in cylindrical orifices. 
The sluice should be inclined (as in Fig. 4), in order to leave 
but a small distance between the orifice and the wheel, which 
win diminish the loss of head produced by the friction of the 



24 UNDEESHOT WHEELS. 

water on the portion of the race M C, through which the water 
reaches the wheel. 



N 



— ^^ 


1 


, ^ ^ 


• 


_lk\ 


"^f!^ 


^ 


' ^ -let 






^f'^^^^' 


^--w^mwy/y/M 


7 



Pig. 4. 



3d. The water leaves the wheel at E F with a velocity v' = 
0.4 V, in the shape of a horizontal band of parallel threads. 
If, in order to flow into the tail race through a section G K, 
where its velocity will be sensibly zero, it had to undergo no 
loss of head, there would be between E F and G K a negative 

/2 2 

V V 

head, the absolute value of which would be ^r-. or 0.16 ^r— : that 

2/ 2^' 

is, the point G would be at a height 0.16 ^ above E, since the 

piezometric levels at E F and G H may be confounded with 
those of the points E and G. It is ik t possible so to arrange 
every part that all loss of head shall be suppressed between 
E F and G K ; but these losses are much diminished by means 
of a plan first recommended by M. Bel anger. The bottom of 
the race, beyond the circular portion A D, presents a slight 
slope, for a distance D I = 1 or 2 metres ; thence it connects 
with the tail race by a line I K, having an inclination from 0°*.07 
of a metre to 0™.10 for each metre in length ; the side- walls are 
prolonged for the same distance, either keeping their planes 
parallel, or very gradually spreading outwards, but never ex- 



UNDERSHOT WHEELS. 25 

ceeding 3 or 4 degrees. The point D is placed at a height 
h' + I, 0.16 ~ (*), or 2.5 h + 0.11 ^- below the level of the 

water in the tail race. From this the following effects take 
place : the water overcomes the difference of level between E 

and G, or the heiarht 0.11 - — , in virtue of its velocitv 0.4 v. 

' ^ 2 ^' .3 

either by a surface counter-slope, or by a sudden change of 

level with a counter-slope, so that the head lost reduces to 

^)^ 
0.05 ^-. 

In many wheels this precaution has been neglected, and the 
level of the tail race has been placed at the same height as the 
point E, and sometimes even below it. We will now show 
that a loss of effective delivery is thus produced. For this pur- 
pose let us see what must, with the above-described arrange- 
ment, be the position of the race relative to the pond, and the 
expression for the effective delivery. In view of simplifying 
this investigation, we will admit that the lines MA, D F, 
slightly inclined, are a portion of the same horizontal. In the 
hypothesis of no loss of head up to B C, the velocity v would 
be due to the height z of the level N in the pond, above the 
highest thread of the fluid vein thrown on the wheel ; but, on 

2 

* The co-efficient o is simply assumed : in replacing- it by unity the 

water would no longer be able to attain the level of the point G, since this 

would require a loss of head which would be null in the interval between 

E F and G K ; consequently the water in the lower race would probably 

drown the floats and impede their motion. It is for the purpose of avoiding 

such an inconvenience that the number in question is taken less than unity ; 

1 ^'-^ 'd' 

the value assumed gives a disposable head expressed by „ g— , or O.OSg-, to 

counterbalance the loss of head of the liquid molecules after escaping from the 
wheel, and to secure for the wheel their free discharge. 



26 UNDERSHOT WHEELS. 

account of the losses of head, we give z a co-efficient of reduc- 
tion, which we will take (for want of precise data) equal to 
0.95 ; that is, we will write 



^ = 0.95 z. 



whence 

z — 



0.95' 2 g 

The distance from the bottom M A to the level K will then be 

1 'y' . 

K -\- ttq^ -^ ; this can also be expressed in another way, thus : 

H+2.5A + |o.l6~ 

by calling H the total head, or the difference in height between 
G and N. Hence 

and consequently 

f^ = 1.057(H + |A), 

a relation which gives v^ for any given head, when the value of 
h has been determined. From this we can deduce z and A + ^, 
which is sufficient to determine the position of the race. The 
dynamic effect T^, from what we have just seen, will be 

A P -^-, or, makmg — ■= 0.4, 



substituting for -x — its value, this relation becomes T. = P 

[ 0.48 X 1.057 ( H + I A ) - 2.1- 1 ^ ] = P (0-507 H - 
0.289 h) 



UNDEKSHOT WHEELS. 27 

T 
The effective delivery p-4f will then be expressed in round 



numbers by 



^^ = 0.50 — 0.3 4; 



for h = O'^.SO and H included between 1 metre and 2 metres, 
it would vary from 0.44 to 0.47. 

Now let us suppose that, without changing the head H, we 
wish to place the level of the tail race below E, or, at most, on 
the same level ; it is plain that we will have to raise the bottom 
of the race. Then v will diminish, and h will have to increase 
in order that the expenditure P may remain the same; for 
these two reasons T^ will diminish, as the above formula (2) 
shows. For example, supposing that the tail race is at the 

height of E, the equation that determines - — will become 

if 

^ + (i£ = ^ + 2-5A' 

whence we derive successively, regard being had to equa- 
tion (2), 



2^ 

3 



0.95 (H + I A) 



Te = P [o.48 X 0.95 (^H + ^ a) - 1.05 a] = P (0.456 H - 



0.375 A), 

and in round numbers 



0.45 - 0.375 ^ 



PH ^.^.^g. 

The data A == 0°'.20 and H = 1 metre would give a effective de- 
livery of 0.38 instead of 0.44; with H = 2 metres, we would 
obtain an effective delivery of 0.41, whilst we had found 0.47- 



28 UNDERSHOT WHEELS. 

There would be a much more marked falling off, if we supposed 
the bottom of the tail race on the same level as the bottom of 
the portion preceding it, as was the manner of constructing the 
race formerly. 

The arrangement of the channel through which the water 
flows off, which we have mentioned as by M. Belanger, can be 
advantageously employed in all systems of hydraulic wheels 
from which the water flows with a sensible velocity, in the 
form of a horizontal current, with parallel threads. The ris- 
ing of the surface of this current taking place beyond the 
wheel, this latter will experience the same action from the 
water, if everything is similarly arranged from the head race 
to the outlet of the wheel. With the canal in question, the 
level rises, instead of remaining the same or falling; we 
then obtain the same action on the wheel with a less head, 
and consequently we can have a greater effect the head remain- 
ing the same. 

However, we see by formula (3) that the effective delivery 
of these wheels never reaches 0.50, in spite of all possible pre- 
veautions ; this system is, then, not comparatively as good as 
those which we are now going to take up. 

5.. Wheels arranged according to Poncelet'^s method. — The 
princij>al cause of loss of work in the undershot wheel with 
plane floats is the sudden change from the velocity v to v'^ 
twice and a half less, which necessarily produces in the liquid 
.a violent disturbance. From this disturbance proceed great 
dnner distortions and a negative work produced by viscosity, 
; all of which diminishes the dynamic effect. The water also 
(possesses a great velocity of exit, which is at best only partially 
turned to account. General Poncelet proposed to avoid these 
inconveniences, continuing, however, to preserve to the wheel 
its special dbar<a€ter, which is rapid motion; that is, he has 



UlSTDEESHOT WHEELS. 29 

endeavored to fulfil for the undershot wheel the two general 
conditions of a good hydraulic motor, viz. : the entrance of the 
water without shock, and its exit without velocity. To this 
end he has contrived the following arrangements : 

The bottom of the head race, which is sensibly horizontal, is 
joined without break to the flume, the profile of which is com- 
posed of a right line of ji^ , followed by a curve. 

The right line forms a slope near the wheel, and its prolon- 
gation would be tangent to the outer circumference of this lat- 
ter ; it ends at the point at which the water begins to enter 
the wheel. The curved portion is composed of a special curve, 
to the shape of which we will return presently, and which 
stops at the point at which it meets the exterior circumference. 
Finally, the floats are set in a cylindrical portion of the race, 
having a development a little greater than the interval beween 
two consecutive floats, and terminated by an abrupt depression ; 
this depression has its summit at the mean level of the water 
down-stream; its object is to facilitate the discharge of the 
water from the wheel. 

The water enters the race under a sluice inclined at an 
angle of from 30 to 45 degrees with the vertical ; the sides of 
the orifice are rounded ofi", so as to avoid the loss of head analo- 
gous to that in cylindrical orifices. 

The floats are set between two rings or shrouds, which pre- 
vent the water from escaping at the sides ; the interior space 
between the rings is somewhat greater than the breadth of the 
orifice opened by the sluice. The fioats are curved ; they in- 
tersect the outer circumference of the crown at an angle of 
about 30 degrees, and are normal to the inner circumference ; 
beyond this, their curvature is a matter of indifference. There 
are ordinarily 36 for wheels of from 3 to 4 metres in diameter, 
and 48 for those from 6 to 7 metres. 



30 IJNDEKSHOT WHEELS. 

The exact theory of this wheel it is almost impossible to 
explain in the present state of science. It is simplified, first, 
by considering the mass of water that enters between two con- 
secutive floats as a simple material point, which, during its 
relative motion in the wheel, would experience no friction. 
We suppose, moreover, that the absolute velocity of this point 
is in the direction of the horizontal that touches the wheel at 
its lowest point, and that the floats are so put on as to be tan- 
gent to the exterior circumference. Now, let v be the absolute 
velocity of the water on striking the wheel, and u the velocity 
on this circumference ; the water possesses relatively to the 
wheel, at the moment of entering the floats, a horizontal velo- 
city V — u^in virtue of which it takes up a motion towards the 
interior of the basin formed by two consecutive floats If we 
liken, during this very short relative motion, the motion of the 
floats to a uniform motion of translation along the horizontal, 
the apparent forces will reduce to zero, so that the small liquid 
mass that we have spoken of will ascend along the floats to a 

(,y ^y 

height — ^ -, on account of its initial relative velocity, which 

t/ 

is gradually destroyed by the action of gravity. Then this 
mass descends, and again takes up the same relative velocity, 
V — u, when on the point of leaving the float ; but this relative 
velocity will be in a direction contrary to the preceding, and, 
consequently, also in a direction contrary to the velocity u of 
the floats. The absolute velocity of the water on leaving will 
then be equal to the difference between u and v —u, or 2u — v ; 

we see that it will be zero if we have u = -v, that is, provided 

that the velocity at the outer circumference of the wheel is half 
that of the water in the supply channel. 



UNDERSHOT WHEELS. 31 

We could thus realize tlie two principal conditions for a 
good wheel. But, as M. Poncelet has observed, such favorable 
circumstances are found by no means in practice. 

The water cannot enter the wheel tangent to its circumfer- 
ence, as we have here supposed. In fact, let us call ds the 
length of an element of this circumference immersed in the 
current, h the breadth of the wheel, ^ the angle formed by ds 
and the relative velocity, w, of the water referred to the wheel ; 
there will enter during a unit of time, through the surface bds, 
a prismatic volume of liquid having for a right section Ms. sin 
/3, and a length v) / in other words, a volume hds.w sin /S, a 
quantity that reduces to zero, for /3=0. Hence w must inter- 
sect the circumference at a certain angle which cannot be zero ; 
besides, we must make it as small as possible, in order that, at 
the point of exit, the relative velocity of the liquid and the 
velocity of the floats may be sensibly opposite, and give a resul- 
tant zero ; on the other hand, it must not be so small as to 
make the entrance of the water difficult or impossible. It is 
in order to reconcile these two contradictory conditions that 
the angle /3 has been fixed at 30 degrees, which is also that made 
by the floats with the exterior circumference, since the threads 
must enter in the direction tangent to the floats. But then 
the absolute velocity of the water is no longer zero on leaving 
the wheel, for its two components are no longer following the 
same right line, but make an angle of 180°— 30° or 150°. 
Now, supposing V — u = u, the resultant v' would have for a 
value 2u cos 75°, or v cos 75°, or finally, 0.259'y; this resultant 
lies furthermore in the direction of the bisecting line of the 
angle between u and v — u; that is to say, it is almost vertical, 
and consequently it is impossible to turn it to account by 
means of a counter-slope; its efiect is destroyed in producing a 
disturbance in the lower portion of the canal, whence there 



82 UNDERSHOT WHEELS. 

results a negative work equal to P — -, P being the expenditure 
of the head. It is then a loss of head expressed by —- , or 

0.067-^. 

As a cause of loss we may still mention the friction of the 
water against the race and against the floats. Another very 
serious objection to the above-mentioned theory is that the 
liquid molecules do not move as though they were entirely 

isolated ; when one of them, having reached the height -^— — <- 

between the floats, is in its descent, another has just entered, 
and it is not clear that no sensible disturbance in the motion 
of the molecules follows. 

On account of all these reasons, experiment indicates only 
an effective mean work of 0.60 for these wheels. Nevertheless, 
the improvement is very great on the old undershot wheels 
with plane floats, whose effective work was scarcely 0.25 or 
0.30, and could never reach the limit 0.50. As to the most 
favorable ratio between the velocities w and v^ experiment 

gives it 0.56 instead of -. 

It has been observed that the straight portion of the race was 
followed by a curve; the following condition determines its 
form. The relative velocity of the water v — u 2X its entrance, 
equal to that which exists at its exit, is also the same as the cir- 
cumference velocity u ; then the direction of the absolute velo- 
city, resulting from these two velocities, is on the line bisecting 
the angle formed by these two, and since the angle between 
the tangent to the exterior circumference and the relative 
velocity is 30 degrees, that between the same tangent and the 
absolute velocity will be 15 degrees. All the threads, then, 



„f->Tir /-\ II, • Trif-» TifjpTiHTi- 




UNDEKSHOT WHEELS. 33 

must make an angle of 15 degrees with tlie circumference. 
To deduce from this the shape of the bottom, let us assume 
that every normal to the bottom is at the same time normal to 
all the threads that it intersects. Let A then (Fig. 5) be the 
point of entrance of a thread, the absolute velocity A 'y of which 
makes an angle of 15 
degrees with the tan- 
gent Ku; the perpen- 
dicular B A B' to Av 
is a normal to the 
bottom of the race. 
At the same time, if 
we draw the radius 
AO,theangleOAB ^^^- '• 

will be equal to uk.v^ as their sides are perpendicular ; hence 
O A B' == 15 degrees. The perpendicular O B' let fall from the 
centre O to the prolongation of B A, has then a constant value 
equal to A O sin 15°. Consequently, all the normals to the 
bottom of the race are tangent to the same circle, having a 
radius O B' ; the curve C B D is then the involute of which this 
circle is the evolute. It is terminated at one end by the cir- 
cumference OA, at the point D, and at the other at a point C, 
such that the normal C E, taken as far as the circumference 
O A, may have a length equal to the thickness of the fluid 
stratum. This thickness, moreover, varies with the head ; it 
must be, according to experiment, from 0°^.20 to 0°^.30 for 
heads below 1™. 50, and may be diminished to about 0^.10 for 
heads of more than 2 metres. 

The water rises in the float to a height ^—- — ^ = - — - , 

^ ^g 4 2^' 

which differs little from - H, calling H the height of the head. 



34 UNDERSHOT WHEELS. 

The distance between the two circumferences that limit the 

float should be at least - H ; in order more certainly to avoid 

the possibilit}^ of the water still possessing any relative velocity 
on reaching the extremity of the floats, which would give rise 
to a spirting of the water to the interior of the wheel, M. Pon- 

celet has advised increasing this advance to - H. 

o 

6. Paddle wheels in an itnconfined current. — These wheels 
are placed in a current whose section has a breadth much 
greater than that of the wheel ; frequently they are supported 
by boats, and are called hanging wheels. It being impossible 
to calculate theoretically the dynamic effect of the current on 
these wheels, we will be satisfied with the following general 
ideas. 

The horizontal force F which the wheel exerts on the liquid 
being constant, its impulse in the unit of time lias numerically 
the same value as its intensity. If, then, owing to this impulse, 
a mass m of water passes, in a second, from the velocity v of 
the current to the velocity u of the wheel, we shall have 

Y = m {v — u)^ 

and consequently the work produced on the wheel in the same 
time, sensibly equal to the dynamic effect of the current, will 
be 

Y u = m u (v — u). 

Moreover, General Poncelet supposes that the mass m must 
be proportional to v, and, furthermore, it is quite natural to 
admit that it is proportional to the area S of the immersed 
portion of the floats. He then places, calling B a constant 
co-efficient and n the weight of a cubic yard of water, 

m = B S ^ — 
9 



UNDERSHOT WHEELS. 35 

whence we have 

g 

This formula has been found quite true by experiment, in 
taking B = 0.8. 

When V is given, the maximum state of F 'i^ corresponds to 
u = v — u, since the sum of these two factors is constant : from 

this we deduce '?^ = - -y, as we found for the two wheels pre- 

viously discussed. Experiment indicates the ratio - = 0.4 as 

being the most suitable; this ratio only changes very slightly 
the theoretical maximum, the value of which is 

0.4:nS'y-^. 

The depth of the floats must be from f to - the length of 

the radius. Flanges placed on the edge, on the side which re- 
ceives the shock of the water, will increase the mutual action. 
The diameter is ordinarily from 4 to 5 metres, and the floats 
are 12 in number. 



BREAST WHEELS, 



7. Wheels set in a circular race^ called hreast wheels. — These 
wheels resemble very closely, in their construction, the under- 
shot wheels noticed in No. 4; but one essential diiference, which 
can produce a great change in their efiective delivery, con- 
sists in the method of introducing the water ; this no longer 
enters the wheel at its lower portion, but at a slight depth only 
below the axle — that is to say, on the side. The questions now 
to be successively examined will be, 1st, the most suitable ve- 
locity of the wheel ; 2d, the method of introducing the water ; 
3d, the situation of the race as regards the upper end and 
tail race ; 4th, the manner of determining the dynamic effect. 
Finally, several practical ideas will be put forth on the subject 
of this wheel. 

{a) To determine the velocity of the wheel. — Let Q be the 
volume that the head expends in a second ; 

h the breadth of the wheel, which is the same as that of the 
sluice ; 

h the depth of the immersed portion of the floats directly 
beneath the axle; 

c the thickness of the floats ; 

C their distance apart, measured between their axes along 
the exterior circumference ; 

R the radius of this circumference ; 

u the velocity of any one of its points. 



BREAST WHEELS. 37 

The water comprised between any two consecutive floats, 
directly beneath the axle, will have for its depth A, for its mean 

breadth C ( 1 — ^pp) — <?, and h for its thickness parallel to the 
axis of the wheel ; its volume is then hhCil — —^ — tt )? ^^^j 



u 



supposing that it expends during one second a number ^^ of 
these volumes, we shall have 

h G 

In practice — ^ and — are always small fractions, the sum of 

which rarely exceeds 0.10 ; we would then have, except a 
slight error, 

Q rrr 0.9 A 5 t^. 

Q being given, we can satisfy this relation by assuming u and 
h arbitrarily, and then determining 5. In this case, we would 

lOQ, 
9A w' 



take h from 0^.15 to 0°^.25, u = V^.SO, and finally, h = 



the expenditure -^ per metre in breadth, expressed by 0.9 h u, 

would then vary from 175 litres to about 270 litres. 

The values just given for h and u may be satisfactorily ac- 
counted for as follows : 

The calculation of the losses of head sustained by the water, 
in its passage from the head race to the tail race, shows, as we 
shall see further on, that all these losses increase with the velo- 
city of the wheel. We should naturally be inclined to make 
this velocity very small, in order to increase the effective deliv- 
ery. But several conditions show that it is not well to allow 



38 BREAST WHEELS. 

the wheel to move too slowly. In fact, we see that if u were 

very small, the product hh = — would be large, and one of 

y u 

the two dimensions h or h would have to be quite large. Now 
a great breadth h would give us a heavy wheel, expensive to 
put up, losing a great deal of work by the friction of the axle ; 
a great value of h would produce inconveniences already men- 
tioned in speaking of the undershot wheel (No. 4). Finally, it is 
well to have the wheel perform, up to a certain point, the func- 
tions of a fly-wheel for the machinery that it sets in motion, 
which is another reason for allowing it to retain a certain 
velocity. The value u = 1™.30 has been given by experiment. 
As regards the depth A, it is well, on account of the unavoida- 
ble play between the wheel and the race, that it should not 
descend much below 0°^.15, in order not to lose too great a pro- 
portion of water. 

But it sometimes happens that, in taking u = 1°^.30, and h 
within the limits above mentioned, we arrive at a great value 
for h, or a value that goes beyond a fixed limit, either on ac- 
count of local circumstances, or through economy in construc- 
tion. We are then obliged to increase A or u. It is not well 
to have h greater than 0^A5 or 0°^.50, and when this limit is 
reached, we must then begin to increase u. 

For example : let Q = 0°^.50. Taking u = 1^.30, h = 0^.20, 

we would have h = — j^= 2°^.14, a value that in general is 

quite admissible. But if, by reason of particular circum- 
stances, we could not exceed a breadth of 0°^.71, about the 
third of 2°^.14:, we should first increase A, bringing this up to 

0^.50 ; we would then find u -^ ^^^ =: 1°^.56. Or else, as the 

9 o/i 



BKEAST WHEELS. 



39 



velocity 1^.56 is not yet very great, we would be content to 
take k = 0°^.40, which would give u = V^M. 

(h.) Method of introducing the water. — As has already been 
said (1^0. 3), in order to diminish as much as possible the loss 
of work produced by the introduction of the water on the 
wheel, we must so arrange matters as to have a small relative 
velocity of the water at its point of entrance, or, when that 
cannot be obtained, the relative velocity must be tangent to 
the first element of the floats, and the water must move to the 
interior of the wheel without shock, merely gliding along the 
solid sides. 

If the wheel be moving slowly — that is, if the velocity at the 
circumference be about 1™.30 per second, we should let the 
water in by the means indicated in the following figure. 




"^^^^^^^^^^^^ 



^^^^^^^^S^^^^5^^^^^J^^^5^5^j^j^ 



Fig. 6. 



The flume A B, constructed of masonry, is prolonged by a 
piece of cast iron B C, called a swan'^s neck^ or guide bucket ; 
AB C is the arc of a circle nearly coincident with the exterior 
circumference of the wheel, leaving the slightest amount of 



40 BREAST WHEELS. 

play. A sluice D, furnished at its upper extremity with a 
small, rounded metallic appendage E, can slide along the 
guide bucket while resting on it ; a system of two racks con- 
nected by cog-wheels allows this sluice to be raised to any con- 
venient point. The metallic appendage E forms the sole of a 
weir over which the water flows to get to the wheel ; this sole 
must be from 0"^.20 to O'^.ST below the level of the pond. 

The sluice being thus very near the wheel, the absolute 
velocity of the water at its entrance, due only to the slight fall 
which takes place in the surface of the pond, will consequently 
be small ; and as the wheel itself moves slowly, the relative 
velocity will be moderate, and the disturbance produced hardly 
perceptible. We see that, in order to attain this end, we should 
reduce the head over the sole of the weir, for in the contrary 
case the velocity of flow would have a greater or less value ; we 
should not, moreover, carry the reduction too far, in order that 
the loss of water between the wheel and flume may not be too 
great. The limits 0°^.20 to 0"^.27 fulfil this double condition 
quite well; they correspond very nearly to the limits 0°^.15 and 
0™.25 above mentioned for the depth h of the current just 
beneath the axle : for a weir without lateral contraction can 
yield about 180 litres per meti-e of breadth, with a head of 
0°^.20 above the sole, and 280 litres with a head of 0°^.27. 

The rounded metallic appendage E has for its object to 
diminish the contraction of the sheet of water flowing over the 
weir, and thus for the same head to yield more water: the head 
and velocity of flow are then less for a given yield, which 
diminishes the disturbance of the liquid on entering. The 
piece E may also serve to direct the vein so as not to intersect 
the exterior circumference of the wheel at too great an angle ; 
this is as it should be, for, all other things being equal, the 
relative velocity of the water increases with the angle in ques- 




BREAST WHEELS. 41 

tion, as the following considerations sliow, altliough they more 
particularly apply to a different case. 

Let us now pass to the case of rapidly moving wheels. The 
velocity u of the wheel at its circumference has been fixed as 
has been shown ; but we still, in order to lessen the loss due to 
the introduction of the water, dispose of the absolute velocity 
V of this latter, in intensity and direction. Let, at the point 
of entrance, M (Fig. 7), M U be the velocity 
u of the wheel, MY the absolute velocity v 
of the water, 7 the angle formed by these two 
right lines. The line M W, equal and paral- 
lel to U Y, will be the velocity w of the water 
relatively to the wheel. The first point is to 
make v^ as small as possible, ^ow we should 
j,j^ Y have w =^ o^ii y were zero and 'y equal to u ; 

but it is not possible for the water to enter 
without any relative velocity, and at an angle zero. One 
thread for which 7 may have the value 0, would be tangent to 
the exterior circumference of the wheel ; but it is evident that 
the other threads, placed in juxtaposition parallel to this, could 
not fulfil the same condition, unless the total thickness of these 
threads taken together were itself zero, which cannot be admit- 
ted in the case of a finite expenditure. We have seen that in 
Poncelet's wheel the angle 7 was taken equal to 15 degrees ; in 
the breast wheel it is generally taken to be 30 degrees, in order 
to somewhat facilitate the introduction of the water. Then, 
since U is a point determined as well as the direction M, the 
smallest possible value of TJ Y = ^iJ would be the perpendicular 

let fall from U on M Y, or u sin 7, or finally -uii y he, taken 

equal to 30° ; v would have for a corresponding value u cos 7, 
or 0.866 V, 7 being 30°. 



4:2 BREAST WHEELS. 

Still these values are not those taken for v and w. It is well, 
in fact, to have the first element of the floats in the direction 
of w, in order to diminish the disturbance of the water within 
the wheel. 'Now, if this first element were perpendicular to 
M Y, it would make an angle of 120° with M IT — that isj with 
the circumference of the wheel, and consequently an angle of 
30° with the radius through M. When this radius, by the act 
of revolving, reaches the vertical, immediately below the axle, 
the first element of the float will still be inclined at an angle 
of 30° with the vertical, and it wo aid be still greater at the 
point of emersion, which would obstruct the motion of the 
wheel. On this account it is better that the relative velocity 
M W should pass through the axis of rotation and be perpen- 
dicular to M U ; the parallelogram of velocities is then repre- 
sented by the rectangle MUY'W', in which M U = 'W, 
M W — ^(?, MY ' = V. The angle / still preserving its value 
of 30°, we have these relations : 

COS 30° ' 

w = u tang 30° = 0.577 u. 
The intensity of v and its angle with u being henceforth de- 
termined, it remains to be seen how in practice these two con- 
ditions can be realized. To obtain them, we will first deter- 

mine the height - — ; we will increase it a little, say by one- 
tenth, in order to compensate approximately for the losses of 
head sustained by the liquid between the head race and tlie 

point of entrance, and 1.1 - — will be the depth of the point of 

2 g 

entrance below the level of the head race. The direction of 
the velocity v will be obtained by drawing through the point 
of entrance, which we have just found (since it is on the ex- 



BREAST WHEELS. 43 

terior circumference of tlie wheel, and on a known horizontal 
line), a line making an angle of 30° with the said circumfer- 
ence. The threads should then be obliged to take this direc- 
tion bj means of a canal of from 0^.50 to 0™.60 in length, the 
last element of which should be tangent to -y, and into which 
the water would flow, either by passing under a sluice, or by 
flowing out without any obstacle, the bottom of this canal 
being then the sole of the weir. 

(c.) Position of the flume as regards the head race and tail 
race. — We have only to repeat here what has already been said 
in discussing the undershot wheel with plane floats. When 
the water possesses, on leaving the wheel, a sensible velocity, 
it is well to place the level of the portion between the floats, 
which is directly under the axle, below the level of the lower 

'2, u^ . 

section, by a quantity equal to - — ^ to being the velocity of 

the wheel at the circumference. The velocity u, having a 

value previously determined, as well as the height h of the 

water between the floats, the situation of the bottom of the 

flume is then also fixed, at least that part below the axle. On 

the up-stream side this bottom has a circular profile, with a 

radius sensibly equal to that of the wheel ; it should be joined 

with the tail race in accordance with M. Belanger's rules, of 

which we have previously spoken (No. 4). 

When the wheel is moving slowly, u being about 1°^.30, 

2 u^ 

^ ^ — gives a depth a little below 0°^.06. However great the 

head may be, the loss or gain of 0°^.06 is of but small import- 
ance, and we would then be able, by economy, to do away with 
the portion of the race beyond the wheel, or at least to do 
without a masonry-lined canal with a rescular section. The 
level of the water between the floats would then coincide with 



4:4: . BREAST WHEELS. 

tliat of the tail race. But in the case of large values of u, es- 
pecially if, at the same time, we have only a slight head, the 

height - - — may afford a sensible gain that we should not 

if 

neglect. 

id.) To calculate the dynamic effect of a head of water 
which sets a hreast wheel in motion, — Calling P the weight of the 
water expended in a second, H the head measured between the 
two portions of the canal supposed to be in a state of rest, we 
know (E"o. 2) that the dynamic effect T^ of the head, during 
each second, has for its value the product of the weight P by 
the height H, diminished by the mean loss of head that the 
liquid molecules sustain by reason of friction in their passage 
from one portion of the canal to the other. Moreover, this loss 
of head is composed of several portions, which we shall now 
proceed to analyze, using the notation adopted (JN^o. 7, a, 
h. c). 

1st. Loss of head letween the head race and the point at 
which the water enters the wheel. — This is reduced by avoiding, 
by means of rounded outlines, contractions followed by a sud- 
den expansion, and by placing the point of entrance as near 
the head race as possible. Nevertheless there is always a cer- 
tain loss : this we may assume at first sight as comprised 

between 0.05 - — and 0.1--- . If there were a canal between 

^ g 2 ^ 

the wheel and the head race, we should have to take the fric- 
tion in this canal into consideration, by a calculation analogous 
to that which will be presently given for the circular flume. 

2d. Loss of head due to the introduction of the water. — After 
what has been said (in No. 3), this loss may be valued at 

w"^ 1 

- — , or at - — (v^ -{- u^ — 2 u V cos y). In the case of slowly 

2^ 2^' ^ -^ 



BREAST WHEELS. 45 

moving wheels it is of slight importance, like all the other 
losses that we consider ; in one moving quickly, if 7 = 30°, 

u 

V = , we shall have 

cos 7 



w u , 2 



tan 7 = 



2y 2^ ' 3 2/ 

When the precaution is taken to introduce the water with a 
relative velocity tangent to the floats, and a surface is presented 
along wliich it can ascend in virtue of this relative velocity, it 
is probable that the action of gravity assists in overcoming this 
velocity, and thus so much less eflect will be consumed in a 
violent disturbance of the water. This consideration justifies 
the use of polygonal floats, such as are shown in (Fig. 6), the 
arrangement of which was contrived by M. Belanger in 1819. 
They are composed of three planes making angles of 45 degrees 
with each other, of which the furthest from the centre is in the 
direction of a radius, the nearest touches the circumference of 
the ring, and the third connects the two others. The planes 
that are fixed to the crown have vacant spaces left between 
them to facilitate the disengagement of the air. The point of 
entrance of the water must, moreover, be lower than the centre 
of the wheel, in order that the water introduced may be 
received upon an ascending inclined plane. 

Data are wanting to estimate the effect produced by the use 
of this kind of float. 

3d. Loss of head produced hy the friction of the water 
against the circular flume. — "We know that the friction of a 
current on its bed, per square yard, is expressed by 0.4 U'^, IT 
being the mean velocity. Moreover, calling Y and W the 
velocities at the surface and at the bottom, it has been found 
that these quantities are connected by the approximate rela- 
tions : 



46 BREAST WHEELS. 



whence 



U = i(y + W), U = 0.80Y; 



u = |w. 



The friction per square yard of bed wonld then be 

0.4. ^W^ or 0.71 W^ 

Now, if L be the length of the circular race, L (5 4- 2 A) will 
be the surface wet, and the entire friction is expressed by O.Tl 
L (J + 2 A) %i^^ since the velocity at the bottom is the same as 
the velocity at the circumference of the wheel. All the points 
of application of the forces that compose this friction moving 
with the velocity u^ their negative work in the unit of time 
will be 0.71 L (J + 2 A) 'i^^ ; we shall obtain the corresponding 
loss of head by dividing by P, or by 1000 hhu^ which gives 

0.00071 — -^ — %—r^ , or else 0.014 ^ , ' ' .—-. 

oh h ^ g 

4th. Loss of head from the point just heneath the axle to the 
tail race. — If the level of the water between the floats, beneath 
the axle, exceeds by a height r) the level of the portion lower 
down, the velocity of the water will increase on account of this 
difference of level. This velocity, sensibly equal to u on leav- 
ing the wheel, will become Vu^ -\- 2 g -n, for the point at which 
the water begins to enter the lower portion. As we have seen 

(Ko. 3), a loss of head >] + - — corresponds to this velocity. 

2 g 

When we adopt the arrangement of the tail race recom- 
mended by M. Belanger (No, 4), y] becomes negative and equal 

to - - — ; the loss then reduces to - ■ — . 
S2g' d2g 

{e) Some practical data. — With regard to the number of 



BEEAST WHEELS. 47 

arms, the same rule here applies as to undershot wheels 
(Ko. 4). 

The number of floats is a multiple of the number of arms ; 
their distance apart may be from once and a third to once and 
a half the head above the top of the weir, in the case of slowly 
moving wheels. In wheels of rapid motion, it will always be 
necessary to take this distance apart a little greater at the por- 
tion of the circumference intercepted by the stratum of water 
that falls upon the wheel. It is not well to place the floats too 
far apart, because some of the threads might fall from quite a 
considerable height before reaching them ; it is also bad to 
place them too close together, because the water could with 
difficulty enter the wheel, and a portion would be thrown off. 

The depth of the floats in the direction of the radius is but 
little greater than 0°^.70. In its normal condition, the interior 
capacity formed by any two consecutive floats should be very 
nearly double the volume of water that they contain : we 
might then take the depth in question equal to 2 A, whenever 
A is not greater than 0°^.35. 

The diameter of the wheel should be at least 3°^.50 ; it is sel- 
dom greater than 6 or 7 metres. The axis of the arbor is 
placed a little above the level of the up-stream portion of the 
canal. 

Breast wheels are suitable for falls of from 1 to 2 metres, or 
even 2°^. 50. Beyond these limits they can still be frequently 
used to advantage. 

When a breast wheel of slow motion is well organized, the 
dynamic effect of the head may approximate quite near to what 
is due to the entire head. General Morin, in some experiments 
on these wheels, found an effective delivery of 0,93, But, 
since the measure of tlie expenditure of water is always some- 
what involved in uncertainty, and consequently the total effect 



48 BREAST WHEELS. 

due to a given head can be but imperfectly ascertained, it will 
be prudent not to count beforehand on an effective delivery, in 
practice, greater than 0.80. 

8. Examjple of calcidations for a rapidly moving Ireast 
wheel. — The wheel in question was experimented on by Gene- 
ral Morin ; it belonged to the foundry at Toulouse. 

The water left the head race under a sluice that was raised 
0°^.147 above the sole, which was 1™.423 below the above-men- 
tioned level. The orifice being prolonged by a very nearly 
horizontal race, the velocity v with which the water reaches the 
wheel will be due to the head at the upper portion of the ori- 
fice, except a co- efficient of correction very nearly 1, which we 
will value at 0.95 ; hence 



V = 0.95 V 2g (1^.423 - 0°^.14T) = 4°^.75. 
The velocity at the circumference of the wheel was u = S'^.OB, 
and the angle made by u and v, at the point of introduction, 
was valued at 30°. It follows directlv from this, that the loss 
of head sustained in bringing the water from the head race to 

the wheel should be ^ of the head (1°^.423 — 0°^.147) or 0^.13 ; 

we can also calculate that produced by the introduction of the 
water, which was expressed by (No. 7, d) 

2 g 2 g\ J 

The level of the water between the floats, just below the 
axle, being at the level in the down-stream portion of the canal, 

we must again count upon a loss equal to - — (ITo. 7, d), or 

2 g 

0"^.48. 

Finally, there is a loss in the circular flume. The depth of 

the water beneath the axle and the breadth of the wheel being 

valued respectively at 0™.20 and 1°^.55, and the length of the 



BKEAST WHEELS. 49 

flume beinff 2°'.50, we find for this loss 0.014. ^ ^^ -. ^ k ^ , 
^ 0.20 X 1.55 2 g^ 

or 0^^.11. 

All these losses together make up a head of 0"^.13 + 0™.34 

+ 0^^.48 + 0^^.11, or of 1°^.06. The head being 1^.72, we see 

that the productive force as calculated would be onlj 

— — zr-f^^ — , that is, 0.38 ; M. Morin found experimentally 0.41, 
1.72 

a number which corresponds to an available head of 

1°^.Y2 X 0.41 = 0°^.705, 

instead of 0"^.66, which the preceding calculation gives. This 

difference, otherwise hardly noticeable, from 0^.045, belongs 

probably to a somewhat greater value of the head lost by the 

introduction of the water ; in fact we have seen that, in certain 

cases, a portion of the relative velocity could be annulled by 

the action of gravity, which would diminish by so much the 

disturbance within the floats, and would give rise to a smaller 

loss of head. 

To increase the effective delivery of this wheel without 

changing its velocity, the following arrangements might have 

2 ij^ 

been made: First, to place the flume at - ^r— , or 0°^.32 lower 

3 2^ 

than its actual position, taking care to arrange the race as 
described by M. Belanger (]^o. 4) — that is, without any sudden 
variation in section, and with a bottom having a moderate 
slope, to its junction with the tail race ; then to raise the point 
of entrance of the water so as to reduce the velocity v to 

-^ or ^ ' ^ = 3°^. 52. The loss from the wheel to the 



cos 30° 0.866 

1 v^ 
tail race would then have been reduced to - - — or to 0°^.16, 

3 2^ 

instead of 0°^.48 ; the loss for the entrance of the water would 



50 BREAST WHEELS. 

also be reduced to the same value 0°^.16 instead of 0^.S4:, which 
would procure a total benefit of 0°^.50. The other losses remain- 
ing sensibly the same, the available head would be 0°^.66 + 0"^.50 
= 1^.16, and the effective delivery would be raised to about 



OYEBSHOT WHEELS. 



9. Wkeels with huckets^ or overshot wheels. — These wheels 
are not, like the preceding, set in a canal. The water is let in 
at the upper portion ; it enters the buckets, which are, as it were, 
Ldsins formed by two consecutive floats, terminated at the sides 
by the annular rings, and closed at the bottom or sole by a con- 
tinuous cylindrical surface concentric with the wheel. The 
questions that the organization of this kind of motor present 
are as follows : 




{a) Introduction of water into the wheel, — Two arrange- 
ments are employed which are represented hereafter (Figs. 8, 
9). In (Fig. 8) the top D of the wheel is placed' a> litde below 



52 OVERSHOT WHEELS. 

the level N N of the pond, at 0^.60 to 0«^.75 lower than that 
level ; the water is led to a point C, situated about 1^.50 up- 
stream, and is delivered directly above the axle, by means of a 
canal A B, or pen trough made of planks, terminated by a very 
thin metallic plate B C, which, being prolonged, would be very 
nearly tangent at D to the exterior circumference. The lateral 
boundaries of the canal ABC are prolonged 1 metre beyond 
the point C, to prevent the water from falling outside of the 
wheel. The water passes over the distance C D in virtue of its 
acquired velocity, and enters the wheel nearly at the top. As 
quite a narrow opening only is left between the soles of the 
buckets, the water that flows in the canal A B C is given the 
form of a thin stratum, by making it pass under a sluice placed 
near A, and which is raised only about O'^.OB or 0"^.10. This 
sluice presents an orifice with rounded edges, so as to avoid the 
eddies consequent to the exit of the liquid threads. As has 
been shown, there is little difference in height between the 
point at which the water enters and tlie level of the head race ; 
consequently tlie water enters the wheel with a slight absolute 
velocity, and if the wheel turn slowly, as it should do, to attain 
a good effective delivery, the relative velocity will itself be 
-slight, as well as the loss of work that it involves. 

The arrangement in (Fig. 9), which has been frequently em- 
iployed, does not appear to be so good ; but we are sometimes 
obliged to make use of it if the pond level is very variable. 
This portion is terminated near the wheel by a wooden shutter 
A B, with openings C C, having vertical faces like those of a 
window-blind ; a movable sluice allows of covering as many 
of these openings as may be requisite, so as to expend only 
the disposable volume of water. The inconvenience of this 
method is, that the water falls through sufficient height into 
rthe buckets to ^ive it a considerable increase of velocity ; the 



OVERSHOT WHEELS. 



53 



disturbance of the water in the wheel thus becomes much 
greater. It tends, moreover, for the same head, to increase the 
diameter of the wheel, which makes it more heavy and expen- 




FiG. 9. 

sive. Besides, the point at which the buckets take a sufficient 
inclination to begin to discharge the water in them is situated 
at a greater height above the lowest point of the wheel, because 
this height is proportional to the diameter; there is thus, then, 
a greater loss of head, seeing that the work of the weight of the 
molecules that have left the buckets, whilst they are falling 
into the race below, is evidently lost to the wheel. 

(h) Shape of the surface of the water in the huckets / velo- 
city of the wheel. — It can be shown that a heavy homogeneous 
liquid cannot be in equilibrio relatively to a system that turns 
uniformly about a horizontal axis. If, however, we admit that 
the relative equilibrium of the water can exist approximately in 
the buckets, which may arise when the disturbance due to the 
entrance of the liquid has nearly ceased, we can determine the 
shape assumed by the free surface as follows : 



54 



OVERSHOT WHEELS. 



Let M (Fig. 10) be a liquid molecule, having a mass m, situ- 
ated at the distance OM. = r from the axis of rotation O ; it is 
in equilibrio relatively with a system which turns around this 
axis with an angular velocity w. This equilibrium exists under 
the action : 1st, of the weight m- g, which acts vertically along 
the line M G ; 2d, of the centrifugal force m w* r, along the 
prolongation M C of O M, an apparent force to be introduced, 
as regards solely a relative equilibrium ; 3d, of the pressures 
produced by the surrounding molecules. We know from the 
principles of hydrostatics that the resultant of the two first 
forces is normal to the surface level (or of equal pressure) which 

passes through M. If, then, M 
be found at the free surface, as 
the pressure there is entirely 
the atmospheric pressure, the 
resultant in question will be 
normal to this surface. Let us 
take MG = mg, M C = m w^ /*, 
the diagonal M B of the paral- 
lelogram M'Gr B C represent the 
resultant of m g and of m w'* r, 
and consequently it is normal to 
the free surface. Moreover, pro- 
longing the vertical O A until it intersects this normal, we 
obtain from the property of similar triangles. 




Fie, 10. 



OA 
MG 



OM 
GB' 



whence 



OA = 



MG^x OM _ mgr _ g 



G B m^' r c^'' 

hence the distance O A is constant, which shows that in a plane 
section perpendicular to the axis all the normals to the free 



OVERSHOT WHEELS. 65 

surface meet at the same point. The profile of the free sur- 
face, if there be relative equilibrium, is then necessarily a circle 
described from the point A as a centre. 

This result proves that, in effect, the relative equilibrium is, 
strictly speaking, impossible ; for, in proportion as the bucket 
leaves its place, the point A not changing position, the free 
surface would have an increasing radius, which is incompatible 
with the hypothesis of relative equilibrium, since in this case 
the form of the free surface ought not to change. The form 
that has been determined is that which the water endeavors to 
assume without being able to preserve it. 

To finish determining the circle which limits the water in a 
given bucket, a circle of which we as yet know only the centre 
A, we must take into consideration the quantity of water that 
the bucket is to hold. To this end, let K be the number of 
buckets filled, h the breadth of the wheel parallel to the axis, 
Q the volume expended by the pond per second. Each bucket 

occupies on the circumference an angle -^^ (expressed in terms 
of an arc of a circle having a radius 1), and as the wheel turns 
with an angular velocity w, — — will represent the number of 
buckets filled in a unit of time. Each bucket then contains a 
volume — ^-, so that the area occupied by the water in the 

cross section of the bucket has for its value -^ — ~. The arc 

D M E will then be determined, since we know its centre and 
the surface D M E F 1 which it must intercept in the given 
profile of the bucket. 

In a certain position I^ E^ F^ of the bucket, the free surface, 
determined in the way just mentioned, just touches the edge of 



56 OVEKSHOT WHEELS. 

the exterior side E' ; this position may be found by trial. As 
soon as the bucket passes it, the water begins to run out ; for 
every position below this, it is clear that the free surface will 
have for a profile a circle having A for a centre and just touch- 
ing the outer edge of the bucket, and which will allow the 
volume of water remaining in the bucket to be determined. 
When the circle in question passes entirely below the profile of 
the bucket, the discharge will be complete. 

In practice, if the question relates to wheels possessing only 

a slow angular velocity, -^ will be so great that the circles de- 
scribed from A as a centre, to limit the surface of the water in 
the buckets, may be assumed as horizontal lines. Example : 
The wheel being four metres in diameter, and having a velocity 

of 1 metre at the circumference, then a ^=- and -^ = 39°^.24r, 

or the distance of the centre A above the axis. 

When an overshot wheel turns rapidly, the distance ~ may 

become so small that the free surface may present a noticeable 
concavity below the horizontal ; thus the more the angular 
velocity increases the less water the bucket can hold in a given 
position, which is easily seen, since the centrifugal force becomes 
greater and greater, and this force tends to throw the water out 
of the bucket. This is an inconvenience attendant upon wheels 
that turn rapidly ; they lose a great deal of water by spilling, 
and consequently yield a smaller effective delivery. 

The losses of head produced by the introduction of the water 
into the buckets, and by the velocit}^ of the water when it 
leaves the wheel, also increase with the angular velocity. We 
will then be led, in order to economize the motive force as 
much as possible, to make the wheel turn very slowly. But 



OVERSHOT WHEELS. 57 

we have already seen, in speaking of breast wheels, that it is 
not w-ell to make a water wheel move very slow, because, in 
order to use up an appreciable volume of the water, it would 
be necessary to establish a machine of immense size. A velo- 
city of from 1 metre to 1°^.50 at the circumference gives good 
results. 

(<?) Breadth of the wheel ; dejpth of the huchets in the direc- 
tion of the radius. — We have said above that, in a well- 
arranged wheel, the water leaves the up-stream portion of the 
troughs by passing under a sluice raised from 0°^.O6 to 0^.10 
above the sole ; which is itself from 0"^.20 to 0°^.25 below the 
level of this portion. If we call 

Q the expenditure per second ; 

h the breadth of the wheel, and of the orifice under the 
sluice ; 

X the height to which the sluice gate is raised ; 

h the depth of the sole below the level of the water in the 
up-stream section ; 

V the velocity with which the water leaves the sluice ; 
the velocity v wdll be due very nearly to the head h — x; and as 
the adjustments are so arranged as to have but little contrac- 
tion, we can place 

Q =. 0.95 Ix V^Tfih^^, 
the co-efficient 0.95 being intended to account, at a rough esti- 
mate, for the loss of head that water always undergoes in any 
movement whatever, and for the contraction that would yet 
partially exist. By making, in this expression, h = 0°^.20, x — 

0"^.06, we deduce -^ = 0"^.095 ; in like manner, for h = 0°^.25, 

X = 0^^.10, we find ~ = 0°^.163 ; that is, with the sluice ar- 
ranged as we have have said, we can expend from 95 to 163 



58 OVERSHOT WHEELS. 

litres per metre of breadth of the wheel. It would be easy to 
expend less than 95 litres, by diminishing K and x a little ; we 
can, when necessary, expend more than 163 litres by inverse 
means. But experience shows that, to be in the best condition, 
the expenditure should be but little more than 100 litres per 
metre of breadth ; for otherwise we might be led either to make 
deep buckets, or to cause the wheel to turn rapidly, which 
would tend to increase the velocity of the water when it enters 
or leaves the wheel, and consequently to diminish the effective 
delivery. 

To show the relation that exists between the depth j^ of the 

buckets in the direction of the radius, and the expense -y- per 

yard in breadth, let us preserve the notation already employed 
in the present number, and furthermore let us call 

K the radius of the wheel ; 

u its velocity at the circumference ; 

C = — =^- the distance of the buckets apart ; c their thick- 
ness. 

The volume of a bucket will be equal to the product of its 
three mean dimensions, viz. : its length 5, its depth ^, and its 

breadth C Tl — —^ ~ ^ I this volume has then for its value, 

p b Q (l — ^= — -\ Moreover it would be well, to retard 

the discharge from the bucket, not to have it more than one- 
third full ; the volume of the water it contains would then be 

-phC (l ""o^ — p)) ^^^i ^s we have previously seen, hj 
—^, there obtains 



OVERSHOT WHEELS. 59 

2*0 



whence, because C = — :^^^— and w = :^ ; 

T-S^^'l^ 2K C> 
As the factor in the parenthesis in the second member differs 
but little from 1, we may simply place 

Q 1 

This equation shows that when -j- is large, one of the factors p 

or u must be so too. For example, if -y- = 0.100 litres, and u 

= 1 metre, we find p = 0™.30. It is desirable that p should 
not much exceed 0°^.30. However, if we had an ample supply 
of water to expend, we might either go beyond this limit, or 
use a faster wheel, or finally fill the buckets more than one- 
third. 

The expenditure per metre of breadth having been fijxed, 
from what precedes, as much as possible below 100 litres per 
second, the breadth of the wheel results naturally from the 
total volume of water to be expended. It is seldom that 
wheels having a greater breadth than 5 metres are con- 
structed. 

(d) Geometrical oxitlhie of the huckets. — The distance of the 
buckets apart is a little greater than their depth ; generally, 
this last dimension is from 0°'.25 to 0'^.28, and the other about 
from 0'^.32 to 0'^.35. Their number must be a multiple of the 
number of arms for facilitating the connections, unless the 
crown and arms are composed of a single piece. 

As to their profile, the annexed outline is frequently made 



60 OVEKSHOT WHEELS. 

use of (Fig. 11). After dividing the exterior circumference 
O A into portions A A\ A' A'^, all equal to the distance of 
the buckets apart, we take A D =:j?, the depth of the buckets 
in the direction of the radius, and describe the circumference 
O D ; a third circumference is then drawn, O B, at equal dis- 
tances from the first two. The radii O A, O A', O A'' . . . 



A! A/ 








Fig. 11. 



being then drawn through the points of division, A B', A' B'', 
. . . will be joined, and we shall then have the profiles A B^ D', 
A^ B^^ D^\ . . . which, excepting the thickness, will be those 
of the buckets. 

Skilful constructors think that, instead of the lines such as 
A B', A' B'', ... we might employ the lines a B', a' B'\ . . 
which produce a certain degree of mutual covering between 
the buckets ; in like manner for the lines B D, B' D', B^^ D'V 
. . . , the inclined right lines B d, W d\ W d'\ . . . , have 
been sometimes substituted. These two changes have the one 
end, that of increasing the depth of the buckets in the direction 
parallel to the circumference, and consequently to retard the 
emptying. They are inconvenient, because they make the 
construction more difficult ; besides, this overlapping A a must 



OVEBSHOT WHEELS. 61 

not be carried to excess, otherwise the remaining free space 
between the point B and the side a B' would perhaps be too 
much diminished. This minimum distance should be a few 
centimetres greater than the height to which the sluice gate is 
raised, in order that the water may enter well into the wheel, 
and not be thrown to the outside. 

When the buckets are made of sheet-iron, the broken pro- 
files just mentioned are replaced bj curved profiles, which 
should differ as little as may be from them. 

Wooden buckets are generally from 15 to 30 millimetres in 
thickness ; the sheet-iron ones are only from 2 to 4 millimetres, 
which increases slightly their capacity, all other things being 
equal. They are limited at the sides by the annular rings, 
which are fastened to the axle by arms, which increase in num- 
ber with the diameter of the wheel. They present a continu- 
ous bottom or sole throughout the entire circumference 
D D' D'^ . . . ; in very large wheels this bottom must be 
sustained by supports at one or two points placed between the 
exterior crowns. We might also use in this case one or two 
intermediate crowns. 

{e) To calculate the dynamic effect of a head that causes an 
overshot wheel to turn. — The two main causes which give rise 
to the losses of head to be subtracted from the entire head, to 
obtain the head that is turned to account, are the relative velo- 
city when the water enters the wheel, and that which it pos- 
sesses at the moment that it falls to the level of the tail race. 

It is almost impossible to obtain an accurate value of the 
first. During the time that a bucket is being filled, the point 
of entrance of the molecules, which come in successively, is 
changing in a continuous manner. The first impinge against 
the solid sides ; those that come after, against those that are 
already in; and thence result phenomena very difiicult to 



62 OVERSHOT WHEELS. 

analyze. The study is greatly simplified by admitting, as we 

did in (JS'o. 3), that the height —- , due to the relative velocity 

t/ 

w of the water at its point of entrance, represents the loss of 
head in question. Besides, if we call v the absolute velocity 
of the water, u the velocity of the wheel, / the angle formed 
by the two velocities ; as -z/; is the third side of the triangle 
formed by -y and u^ we shall have 

w^ = vj" ■]- v"^ — ^ uv COS y. 

In reality, the impinging of the water on the wheel takes place 
at different points along the depth of the bucket. Recollecting 
now that the radius of the wheel is great compared with the 
thickness of the shrouding of the buckets, this will not mate- 
rially affect u ; but to determine v and 7, it would be well per- 
haps to suppose the point of entrance, not at the exterior cir- 
cumference, but at the middle of the depth of the buckets. 

Let us pass to the second loss. Let a molecule of the mass 
771 leave the wheel at a height z above the tail race. This 
molecule, having only an insensible relative motion in the 
bucket, possesses, at the moment that it leaves it, the velocity 
u of the wheel, and at the moment it reaches the level of the 
tail race it has a velocity v' equal to V u^ -\- 'i g z. Then it 
gradually loses all its velocity while moving in this portion, 
without its piezometric level changing (for we suppose the free 
surface horizontal in this portion) ; it undergoes then a loss of 
head equal to 

For all the molecules composing the weight P expended in a 
second, there will be a mean loss expressed by p- 2 m ^ 



OVEKSHOT WHEELS. 63 

3 3 -f 

f — + z\ or else by - — + ^ ^ m g 3, the sum 2 including all 

the molecules. This will be the second height to be subtracted 
from the total height of the head ; it is composed of two terms, 
of which the first is at once given, and it only remains to be 

seen how we can calculate the term — 2 m p' s, which expresses 

the special effect of the emptying of the buckets. 

The quantity ^ ^ m g z is nothing more than the mean 

height comprised between the point of exit of a molecule and 
the level of the tail race ; as the circumstances of all the buck- 
ets are exactly the same, it is evidently sufficient to seek this 
mean for the molecules contained in one bucket. To this end, 
we will first determine, as stated above (J), the positions of the 
bucket at which the emptying begins and ends, and, for a cer- 
tain number of intermediate positions, we will ascertain the 
amount of water that remains in the bucket. Let then 

G be the height of the outer edge of the bucket above the 
level of the tail race when the emptying begins, and let the 
bucket, still full, hold the volume of water q^ ; 

c' the analogous height when the emptying has just ended ; 

y the distance that this same edge has descended whilst the 
volume of water g^ was being reduced to g. 

During an infinitely small displacement of the wheel, to 
which the descent d y corresponds, an infinitely small volume 
— dqi^ emptied out, which falls into the tail race from a 
height c — y ; the mean height of the outflow will then be 

— / {^ ~y) d 9.' ^ow integrating by parts there obtains 

2^0 

f(p-y)dq = q{c-y)-{-fqdy, 



64 OVERSHOT WHEELS. 

and, observing that y =z c — & and y =^ o correspond to the 
limit q=z and q = q^^ 

f^'{G-y)dq = q,G+f^_^,qdy = q,c~f^~^qdy; 

1 \ /*G — g' 

the mean sought is then -^^mg z =^ c / qdy. 

/^ ^' 
qdy will be 

effected by Simpson's method, for want of a strict analysis, since 
we have the means of determining the value of q correspond- 
ing to a given value of y. Were we satisfied with a greater or 
less approximation, but generally one sufficient, we could, 

under the signy, replace the variable q by the mean - q^ of its 

extreme values ; we should then find, 

^lmgz = G--{G-G') = -{G + c'). 

To the losses already determined we must still add one for 
bringing the water from the upper portion of the canal to the 
wheel. As we have seen in examining other wheels, it will be 
very slight if a good arrangement be adopted ; its value would 

then be 0.1 - — . 

2 9 

{f) PraGtical suggestions. — The liquid molecules taking, 
during their fall which follows the discharge, a velocity sensi- 
bly vertical, there are scarcely any means for turning this velo- 
city to account by a counter-slope, and consequently the down- 
stream level should just graze the bottom of the wheel. For 
wheels that turn rapidly, it would perhaps be advantageous to 
set them in a mill race, which would only allow the water to 
escape at the lower portion, and with a velocity nearly horizon- 
tal ; we should take care to furnish each bucket with a valve, 



OVERSHOT WHEELS. 65 

placed near the bottom, and opening from without inwards, to 
allow the air to enter when the water runs out. It would then 
be possible to diminish greatly the loss of head occasioned by 
the velocity that the water possesses on leaving the wheel ; but, 
on the other hand, we would increase, in no small degree, the 
expense of erecting the wheel, and very likely also that of 
keeping it in repair. 

Overshot wheels answer very well for heads from 4 to 6 
metres ; less than 3 metres the breast- wheel is to be preferred. 
Besides, as their diameter is nearly equal to the height of the 
water fall, their employment would become practically impos- 
sible for very high falls. 

Experience shows that with an overshot wheel, well set up 
and moving slowly, the productive force of the head of water 
may rise as high as 0.80, and sometimes even more. But, in 
wheels that turn rapidly, it sometimes falls as low as 0.40. 



TUB WHEELS. 



§ III, Water Wheels with Vertical Axles. 

10. Old-fashioned spoon or tiib wheels. — The paddles of 
spoon-wheels are of slightly concave form in the direction of 
their length and breadtli. They are arranged around a vertical 
axle, and receive in an almost horizontal direction the shock of 
a fluid vein which, leaving a reservoir with a great head, is led 
near the wheel by a wooden trough. To obtain a greater 
action, the water is made to strike against the concave side of 
the paddles. 

Let us call 

V the absolute velocity, supposed to be horizontal and per- 
pendicular to the paddle struck, of the vein which strikes the 
wheel ; 

w the section of this vein ; 

u the velocity of the paddles at the point at which they re- 
ceive the shock ; 

n weight of the cubic metre of water. 

The relative velocity of the water and paddles will be hori- 
zontal and equal iov — u ; then, if this phenomenon be assimi- 
lated to that of a fluid vein impinging against a plane, the 

force exerted on the wheel will be n w ^ 1- ; the work 

g 

which this force performs on the wheel in the unit of time will 
be expressed by n w — ^^ L^ a quantity sensibly equal to the 



TUE WHEELS. 



6Y 



dynamic effect of the head (Xo. 2). This expression varies 
with u^ and reaches its maximum for u = -v; this maximum 

o 

4: V^ 

is n w — — , or, seeing that n u v gives the weight P expended 
27 g 

per second, — - P^-— . The heiacht - — can only be a fraction 

^ ' 27 2 ^ 2 ^ -^ 

of the head H; hence, the dynamic effect is found below 

8 8 

— P H, and the effective delivery below — , or about 0.30. 

This number, moreover, cannot be considered as strictly cor- 
rect, because the imperfectness of the theories relating to the 
resistance of fluids has caused us to give a more or less approxi- 
mate value for the reciprocal action of the water and the wheel ; 
however, experience confirms the result of calculation, at least 
inasmuch as it indicates for this class of motors an effective 
delivery that is always very small, varying from 0.16 to 0.33. 

The tub-wheel does not give a much better result. A verti- 
cal cylindrical well made of masonry receives the water from 
the head race through a conducting cliannel A (Fig. 12), whose 
sides incline towards 
each other with an in- 
clination of about - as 
5 

they recede from the 
well ; that is to say, 
that the sides make an 
angle of 11 or 12 de- 
grees with each other. 
One side is also tangent ta the circumference of the well. The 
wheel, whose axis coincides with that of the well, consists of a 
certain number of paddles regularly distributed around a ver- 




FiG. 12. 



68 TUB WHEELS. 

tical axle. The liorizontal section of the paddles presents a 
slightly curved form, having its concavity towards the side 
from which the action of the water comes ; cut by a cylinder 
concentric with the well, they would give inclined lines more 
or less like arcs of helices. The working of the machine is 
easily understood : the water comes through the trough A with 
considerable velocity, endeavors to circulate all around the 
well, and, meeting the paddles in its road, obliges them to turn, 
as well as the axle that supports them. At the same time, the 
water obeys the law of gravity, passes through the wheel by 
means of the free space between the paddles, and falls into the 
tail race, which ought to be a little lower. We see that the 
water must undergo a good deal of disturbance in entering the 
wheel, and, moreover, that it acts upon the latter for too short 
a time to entirely lose its relative velocity. Also the effective 
delivery, sometimes very slight and about 0.15, never exceeds 
0.40. 

We will pass over these primitive machines in order to study 
others more perfect 



TURBINES 



11. Of turbines. — Tlie principle of reaction wheels, sncli as 
are ordinarily mentioned in Treatises of Phjsics, has long been 
known ; but it appears that it was only towards the beginning 
of the last century that the idea was entertained of making use 
of and applying it to the construction of water wheels with a 
certain power. Segner, a professor at Gottingen, and more 
recently Euler, in 1752, made them the object of their 
researches. 

In 1754 Euler constructed another machine, still founded on 
the principle of reaction wheels, but differing from them in 
several important arrangements ; this machine offers the most 
striking resemblance to a powerful wheel now in use, called in 
the arts Fontaine's turbine, from the name of the skilful con- 
structor, who has set up a great many within late years, besides 
extending and completing, in the details of their application, 
the idea first advanced by Euler. It appears that this kind of 
wheel was not much used until towards 1824, the period at 
which the question was again studied by M. Burdin, engineer 
in chief of mines, who constructed a similar machine which he 
called a reaction turbine. In the years following, M. Fourney- 
ron, inspired by the ideas of M. Burdin, established some tur- 
bines in which he introduced marked improvements. Since 
that time turbines have greatly spread and multiplied ; and 
there exists a great number of models which differ more or less 
from each other. 



70 TURBINES. 

Without undertaking to follow up more completely the his- 
tory of the changes successively undergone by this kind of 
wheel, we shall brieiiy describe the three principal classes into 
which the turbines now in use may be divided ; then we will 
give a general theory for them, and finally we will mention 
some details to which a particular interest is attached. 

12. Fourneyron's turhine. — The essential parts of this tur- 
bine are represented in (Fig. 13). 

The water from the head race A descends into the tail race 
B by following a tube, with a circular horizontal section, of 
which C D is the upper opening. This tube, which is perma- 
nently fixed, rests on supports of timber or masonry; it is pro- 
longed by another circular cylinder of cast iron E G I F, mov- 
able vertically, which can be lowered more or less, by means to 
be presently explained. The bottom K K^ K^' \J' \J L of the 
tube is joined to a hollow cylinder or pipe ahcd, supported at 
its upper end ; this pipe is moreover intended to keep the ver- 
tical shaft ^yfrom contact with the water, a motion of rotation 
being given to the shaft by the flow of the water due to its 
head. In fact, we see that if the water reached the axle, it 
would be necessary, in order to avoid leakage, to make this 
latter pass through tightly closed packings, which would occa- 
sion friction, independently of that caused by the contact of 
the fluid. The arbor, as well as the pipe, are, moreover, con- 
centric with the tub. 

Between the bottom G I of the cylindrical sluice E G I F and 
the annular plate K L, there is an opening G K, I L, entirely 
around the perimeter of the bottom, through which the w^ater 
can flow. But as it is important, as we shall see, that the 
threads should not flow in any direction whatever, they are 
guided in their exit by a certain number of cylindrical par- 
titions with vertical generatrices, which are supported by the 



TTIRBrNES. 



71 




^^^^^?:^^^^^^^^^^^^^^^^^^^^^^>;^^?^^^>^ 



J^<^ 




wzm/ 



Fig. 13. 



72 TURBINES. 

plate K L, and of which the arrangement is sufficiently well 
indicated in the horizontal section ; amongst these directing 
partitions or guides, some, such as g A, are joined to the sides 
K^ K^', \J \J' ; others, such as i ^, are shorter, in order to avoid 
too great a proximity in the extremities of the partitions 
towards the axle. 

With regard to the opening G K, IL, the turbine proper 
is included between the two annular plates or crowns 
SRMN, YTPQ; these plates are connected together by 
means of the floats of the turbine, which are cylindrical 
surfaces with vertical generatrices, giving in horizontal sec- 
tion a series of curves, such as I m, jp ^, &c. ; the lower plate 
is further connected with the arbor by a surface of revolu- 
tion T a ^ Pj bolted to it, so as to form a perfectly solid whole. 
The axle rests on a pivot at its lower end ; a lever y ^, 
moved by a rod ^ e, which ends at a point easily reached, 
allows this pivot to be raised a very little, when the wear 
and tear of the rubbing surfaces has produced a slight settling 
of the axle. 

To see how the action of the water sets the machine in 
motion, let us suppose first that the arbor is made fast : then 
the liquid threads, leaving the well through the directing par- 
titions, will strike against the concavity of the floats ; they will 
thus exert a greater pressure on the concave than on the con- 
vex portion of a channel such as Zm^^, formed of two con- 
secutive floats, first by virtue of the shock, and secondly the 
curved path that they are obliged to describe. Hence there 
would result a series of forces whose moments, relative to the 
axis, would all tend to turn the system of the floats in the 
direction of the arrow-head indicated in the horizontal section ; 
thus there will be produced effectively a rotation in the direc- 
tion indicated, if the axle be allowed to turn, even in opposing 



TURBINES. 73 

it by a resistance of which the moment should be inferior to 
the entire moment of the motive forces. 

To diminish as much as possible the loss of head experienced 
by the liquid molecules during their passage from the head 
race to the wheel, we should take care : 1st, to give the opening 
CD a sufficiently great diameter and to round off its edges; 
2d, to furnish the circular sluice with wooden appendages G G', 
II', placed at the lower portion, and having their edges rounded 
off, as shown in the figure. We s^iall thus sufficiently avoid 
the whirls and eddies caused by the successive contractions of 
the threads of water. The wooden packing, moreover, is not 
continuous ; it is composed of a series of pieces, each occupying 
the free space between two consecutive partitions, so that the 
sluice can be lowered without any hindrance as far as the bot- 
tom K K' L L'. 

We shall see, in considering the general theory of turbines, 
how the other conditions essential to a good hydraulic motor 
are fulfilled. 

13. Fontaine^ s turbine. — Fig. 14 shows a general section of 
this machine by a vertical plane. A pillar or vertical metallic 
support A B is set as firmly as possible into the masonry form- 
ing the bottom of the tail race ; this supports at its upper end 
A a hollow cast-iron axle G D E F, which surrounds it ; this 
axle is prolonged above by a solid one, upon which is the mech- 
anism for transmitting the motion. A screw and nut C allows 
the position of the axle to be regulated in a vertical direction. 
Nearly at the level of the water in the tail race (or, if desirable, 
below it) is placed the turbine H I K L M J^ P, permanently 
fastened to the bottom of the hollow axle ; it is comprised 
between two surfaces of revolution concentric with the vertical 
axis of the system ; these surfaces having H K and 1 L for 
meridian lines ; in the intermediate space are placed the floats, 



u 



TURBINES. 




y,— iL 



TURBINES. 75 

which receive the action of the water, and at the same time 
strengthen the two surfaces. The water comes from the head 
race a to the floats by flowing through a series of distributing 
canals, of which the quadrilaterals Q E, H I, S T M N represent 
the sections. These channels are distributed continuously over 
an annular space, directly over the floats ; they are limited 
laterally by the surfaces QHTN, RISM; the space between 
these surfaces, moreover, remains free, except the volume occu- 
pied by the directing partitions, which divide it into a certain 
number of inclined channels, in which the liquid threads move 
with a determinate figure and direction. 

To give a clear idea of the shape of the directing partitions 
and floats, let us suppose a section made by a cylinder or a 
cone, concentric with the axis of the system, passing through 
the middle of the spaces Q R, H I, K L, and this section deve- 
loped on a plane. The developed section of the directing sec- 
tions will give a series of curves such as c d^ ef^ . . . comprised 
in a straight or curved row ; in like manner, for the floats of 
the turbine, we shall obtain the curves dg^fh^ . . . also com- 
prised in another roAv. These curves having been drawn con- 
formably to rules which we shall consider further on, let us 
suppose reconstructed the cylinder or cone that had been devel- 
oped, and let us conceive the warped surfaces generated by a 
right line moving along the axis aud on each of the curves in 
question successively ; we shall in this way have determined 
the surfaces of the partitions and floats. 

It is deemed unnecessary to describe the arrangement for the 
water from the head race to flow into the tail race in no other 
way than through the channels formed by the directing parti- 
tions ; in this respect the figure gives sufficiently clear indica- 
tions. 

We can, as in the case of Fourneyron's turbine (ISTo. 12), ac- 



76 



TTJEBINES. 



count for the direction in which the machine should turn on 
account of the action of the water ; and whicli is that of the 
arrow-head drawn below the development of the floats. 

14. Ko&cUin's turbine. — Koecklin's turbine, of which the 
entire arrangement was first imagined by a mechanic named 
Jonval, does not differ essentially from Fontaine's turbine, 
either in the arrangement of the floats and directing partitions, 
or the mode of action of the water. The most noticeable dif- 




FlG. 15. 



ference consists in this, that the turbine is above the level of 
the water in the tail race, as shown in (Fig. 15), a vertical sec- 
tion of the apparatus. The directing partitions, set in an 



TTJEBINES. 77 

ariTuilar space, of which the trapezoids Q E K I, ST M ]^, 
indicate sections, are fastened to a kind of cast-iron socket, 
which enjbraces the axle A B, without, however, forming part 
of it, or pressing it hard ; they form a set of inclined channels, 
through which the water from the head race flows and reaches 
the floats of the turbine, placed immediately below, as in Fon- 
taine's turbine. These floats are fastened to another socket, 
which is bolted to the axle ; thej occupy the annular space 
H I K L, M :N' O P. 

The inclined channels included between two consecutive 
floats or partitions are limited on the outside by a fixed tub of 
cast-iron, resting on the edges of a well of masonry ; it forms a 
surface of revolution about the axis A B, having Q H K D for 
a meridian section. At the bottom of this tub are found a cer- 
tain number of arms which support a centre piece, on which is 
placed the pivot of the revolving shaft. 

The water having left the turbine, by the apertures K L, 
O P, flows into the tail race by descending through the mason- 
ry well, and then passing into an opening which we may con- 
tract, or, if need be, close at will, by means of a sluice Y. 

The situation of the turbine above the level of the water in 
the tail race allows it to be easily emptied, and herein lies its 
principal advantage ; for this purpose we have merely to leave 
the sluice of the tail race open, and to prevent the water reach- 
ing the distributing channels Q K H I, S T M ^N". We can then 
visit the machine and make the necessary repairs. Besides, 
the height included between the horizontal plane K L O P and 
the level of the water in the tail race should not be considered 
as a loss of head, because it belongs to a diminution of pressure 
on the water which leaves the turbine, and we shall see by the 
general theory, now to be investigated, that this causes an 
exact compensation. 



78 TUEBrNES. 

15. Theory of the three jpreceding turbines. — A complete 
theory of the turbines that we have just summarily described 
ought to include first the solution of the following general 
problem. Having given all the dimensions of a turbine, its 
position as regards the head and tail races, and finally its angu- 
lar velocity, to determine the volume of water that it expends, 
and the dynamic effect of the head, we should then seek the 
conditions necessary to make the effective delivery for a given 
head and expenditure of water a maximum. 

But we shall not treat the question in such general terms. 
In order to simplify the researches with which we are to be 
employed, we shall allow in what follows that all the dimen- 
sions have been chosen and the arrangements made, so that the 
1 1 turbine may fulfil in the best possible waj^ the conditions for a 

good hydraulic motor. Then, from the pond to the exit from 
the directing partitions, care will have been taken to avoid 
contractions and sudden changes in direction of the threads ; 
to have smooth and rounded surfaces in contact with the water 
that flows through, in order that, in this first portion of its 
passage, it may meet with no sensible loss of head. At the 
point of entrance into the turbine, the water possesses a certain 
relative velocity ; matters will be so arranged that this velocity 
shall be directed tangentially to the first elements of the floats, 
in order that no shock or violent disturbance may follow. This 
is a condition that it is possible to fulfil by choosing a suitable 
velocity for the wheel, as well as proper directions for the floats 
and partitions where they join. Finally, as the water leaves 
the turbine in every direction about a circumference, and as it 
is hardly possible to prevent the absolute velocity which it 
then possesses being used up as a dead loss in producing eddies 
in the tail race, we shall suppose that we have taken care to 
make this velocity small. All this combination of circum- 



TURBINES. 79 

stances will considerably simplify our calculations, by allowing 
us to neglect in them, without a very sensible error, the differ- 
ent losses of head experienced by the water up to the point of 
its exit from the turbine, a loss of which the analytical expres- 
sion, more or less complicated, would overload our formulas 
and make them much less manageable. Only, it should be 
understood that our results will be exclusively applicable to the 
case in which the machine works according to the conditions 
for a maximum effective delivery. 

Thus granted, let us call 

V the absolute velocity of the water when it leaves the 
directing partitions and enters the turbine ; 

u its impulsive velocity and w its relative velocity, at the 
same point, with respect to the turbine taken for a system of 
comparison ; 

v\ u' and w' the three analogous velocities for the point at 
which the water leaves the wheel ; 

^ and j^' corresponding pressures at these two points ; 
Pa, the atmospheric pressure ; 

r and r' the distances of the same two points from the axis 
of rotation of the system^ ; 

H the height of the head, measured between the level of the 

(*) These two distances are often equal in the Fontaine and Koecklin tnr- 
bines ; but they should differ in a marked degree in Foumeyron's turbine. 
Moreover, it should be observed that, in the Fontaine and Koecklin turbines, 
the outlets of the water, from the gnide curves, have a certain dimension per- 
pendicular to the axis of rotation ; the lengths r and ?'', be it well understood, 
should be referred to mean points of these outlets. Thus in Fig. 15, for 
example, r should be the mean of U M and UN; r', in the same way, should 

V O + U' P 

be equal to ^ . M N and O P should also be so taken as to be small 

relatively to r and r', in order that the consideration of a single thread of 
water only may not cause an appreciable error. 



80 TURBINES. 

water in the pond and tail race, supposed to be sensibly at 
rest; 

A the depth, positive or negative, of the point of entrance of 
the water into the interior of the tnrbine, below the level of 
the tail race ; 

h! the height the water descends during its motion in the 
interior of the turbine, a quantity equal to zero when M. Four- 
neyron's arrangements are adopted ; 

n weight of the cubic metre of water. 

N'ow, all the intervals between two consecutive directing 
partitions being considered as a first system of curved channels, 
and all the intervals between two consecutive floats as a second 
system. For the first system of these channels, represent by, 

/3 the acute angle under which they cut the plane of the 
orifices that terminate them at the distance ^ from the axis; 
this angle /3 is also that at which the circumference 2 •n' r is cut 
by the partitions ; 

h the depth or breadth of the orifices of which we have just been 
speaking, measured perpendicularly to the circumference '^ rt r. 

For the second system : 

^ the angle made by the plane of the orifices of entrance 
with the direction of the floats, or, what amounts to the same 
thing, the angle of these floats with the circumference 2 "jt r, 
to which we will give, moreover, a direction opposed to the 
velocity u^ the direction of the float being taken in that of the 
relative motion of the water ; 

7 the acute angle at which the channels cut their orifices of 
exit, or, in other words, the angle of the floats with the cir- 
cumference ^ti t' about which the above-mentioned orifices are 
distributed ; 

y the depth or breadth of the orifices of exit, measured per- 
pendicularly to the circumference 2 -n- r' . 



TTEBINES. 81 

The question now is to establish the relations between all 
these quantities, under the supposition that the conditions of 
the maximum effective delivery are satisfied. For that pur- 
pose we shall first follow the motion of a molecule of water 
along its path between the two races, and write out the equa- 
tions furnished by Bernouilli's theorem. 

Between a point of departure taken in the pond, where there 
is no sensible velocity, and the point of exit at the extremity 
of the directing partitions, there is a head expressed by H -f A 

H — — — — ; the velocity being -y, we have then, under the 

supposition of a loss of head that can be disregarded between 
the two points in question, 



2^(H + A + ^i^-) (1) 



The water then moves along the floats of the turbine with a 
velocity at first equal to id and afterwards to %o' : in this second 
period, if we call w the angular velocity of the machine, we 
know, applying Bernouilli's theorem to the relative motion, 

there must be added to the real head h' + ~ — ^-, a fictitious 

2 /2 2 2 '2 2 

head , or — ; still, nes-lectinff the losses 

2 ^ ^ g ^ ^ 

of head, which is approximately admissible when the water 
enters with a relative velocity tangent to the floats, we shall 
then have to place 

w" -w' z^'ig {h' -f -2-^^) j^ u'' -u' . . . . (2) 

When the turbine is immersed and h is positive, the point of 
exit of the water is found at a depth h -\- h' below the level of 
the tail race ; besides, as, for the maximum effective delivery, 
the water must go out with a slight absolute velocity, we can 



82 TUEBINES. 

without material error admit that the pressure varies, in the 
tail race, according to the law of hydrostatics, which gives 

^ + A + A' = 1^ . . . . (8) 

This relation is also true in Koecklin's turbine, although 
h -\- h' becomes negative, provided that the well placed below 
the turbine and the orifice by which it communicates with the 
tail race are sufiiciently large ; for then the water will take up 
in it but a slight velocity and can there be considered as in 
equilibrio ; the pressure p' would then be less than p^ hy a 
quantity represented by the depth — (A + A'), as equation (3) 
shows. We could also preserve it, if the lower plane of the 
turbine, constructed according to one of the first two systems, 
were on the same level as the water in the tail race : we would 
then, in fact, have j^^ =i?' and h + h' =z c>, the quau titles h and 
h' being with contrary signs, or else both equal to zero, accord- 
ing to whether we are considering Fontaine's or Fourneyron's 
turbine. 

The incompressibility of water will furnish us with the 
fourth equation, showing that the volume of water that has 
flowed between the directing partitions is equal to that which 
leaves the turbine. The distributing orifices, left free between 
the partitions, occupy a total development 2 it r (excepting the 
slight thickness of the partitions) and a breadth J, from whence 
there results a surface 2 -tt 5 ^ y as they are cut by liquid threads 
moving with the velocity v^ at an angle /?, we have for the first 
expression of the volume Q expended in a unit of time 

Q r= 2 -TT ^ 7^ sin /3. 'y. 
In like manner, the orifices of exit at the extremity of the 
floats have a total development 2 * 7*^, a breadth h\ a surface 
2 'Tf 6' t'^ and they are cut by the threads moving with a rela- 
tive velocity w' at an angle y' ; then 

Q = 2 -TT 5^ T*' sin 7. w\ 



TTTRBINES. 



83 



Strictly speaking, on account of the thickness of the floats 
or partitions, these two expressions for the value of Q ought to 

undergo a slight relative reduction of— or -— ; but in all cases, 

the reduction being the same for both, we shall have by the 



equality of the values of 



Q 



5 7"y sin /3 = 5 V w* sin 7 (4) 

The three following relations are in a certain degree geo- 
metrical. 

Let us represent (Fig. 16) a float B C and a directing par- 
tition A B ; a liquid mole- 
cule having followed the 
path A B arrives at B with 
an absolute velocity 'y, and a 
velocity w relatively to the 
turbine which itself, at the 
point B, possesses the velo- 
city u. This last being wliat 

is called the propelling velocity, we know that v is the diagonal 
of the parallelogram constructed on u and w / and as the angle 
between v and %b is exactly /?, the triangle BUY will give 




Pig. 16. 



UY r=BU-fBY-2BU.BYcos/3, 

that is, 

^0^ = t^* -f- -y^ — ^ uv cos /3 (5) 

In like manner the liquid molecule, after having traversed, 
relatively to the turbine, the path B C, arrives at C with the 
velocity w' ^ which, taken as a component with the propelling 
velocity u\ gives the absolute velocity n' ; then the angle 7 
being the supplement of that made by u' and w' ^ we shall have 

"^iu' w' 0,0% y (6) ^ \^p' 



ro'' = U" + W' 



84: TURBINES. 

On the other hand, the velocities lo and u' belong to two points 
of the turbine situated respectively at the distances r and r' of 

the axis of rotation, we have then — = — „ or 

r r 

to' r =^ ur' (7) 

There still remains to express two conditions necessary for 

obtaining the best effective delivery. It is necessary first, at 

the point B, that w be directed tangentially to the floats B C 

without which there would be a sudden change of relative 

velocity, whence would result disturbance and a loss of head 

that we have not considered. Now the angle between w and u 

is the supplement of ^, hence the triangle BUY gives 

B U sin B Y U _ sin (B U Y + Y B U) 



or 



It is then necessary that the absolute velocity v' possessed by 
the water on leaving the turbine should be very slight, since 

- — enters in the loss of head (No. 3) : this condition is suf- 
2 g 

ficiently satisfied by taking the angle y small, and placing 

u' = w' ; .... (9) 

for then the parallelogram C U^ Y' W is changed into a lozenge, 

very obtuse at one angle and very acute at the other, and the 

diagonal joining the obtuse vertices is short ; in other words, 

the velocities u' and w' are equal, and almost directly opposed, 

which makes their resultant very small. 

We have thus obtained, in all, nine equations between six- 
teen variable quantities in a turbine, namely — 

six velocities t^, v^ io, u\ v\ w\ 

two pressures p, p\ 



BY" 


" sin B U Y 


sin B U Y 




u _ sin {& + /3) 
V ~ sin ^ 


. . . . (8) 



TUEBmES. bo 

,, r I 

two ratios — r, ttj 

three altitudes H, A, h\ 

three angles /3, 7, ^. 
These equations will serve us for solving two distinct problems, 
which may be thus stated : 1st, having given a turbine and all 

r l 

its dimensions (that is to say, the eight quantities /S, 7, ^, — >■,— 775 

H, A, A'), to show the conditions these dimensions must satisfy, 
in order that the turbine may work with the maximum etfective 
delivery — that is to say, so that the nine above equations may 
obtain ; and, under the supposition that these conditions are 
fulfilled, to show the most suitable velocity of the turbine, as 
well as the expenditure of water corresponding to this velocity, 
its effective delivery, and its dynamic effect ; 2d, having given 
the expenditure and the height of a head, to establish under 
this head a turbine with the best conditions. 

The first question involves eight unknown quantities, which 
are u^ v, w, u', v\ w',p^p' / the elimination of these unknown 
quantities between the nine equations will then give an equa- 
tion of condition to be satisfied by the dimensions of the ma- 
chine — an equation to which we shall have to add two others 
in order to show that the pressures j? and^' are essentially posi- 
tive. The following calculation has for its object to bring out 
these three conditions, and, at the same time, to give the value 
of the unknown quantities. 

Adding equations (1), (2), and (5), member to member, there 
obtains 

w'' = 2g(K + h + h' ^2^^-^ + u''-2uv cos ^, 

or, considering (3) and (9), 

-w -y cos /3 = ^ H . . . . (10) 



0(5 TUBBINES. 

The combination of equations (4) and (9) readily give 

h V r s>in li = ¥ u' r^ sin y • . . . . (11) 
multiplying equations (7), (10), and (11), member by member, 
there obtains 

v^ h t" sin /3 cos /3 = ^ H. h' r'"^ sin y ; 

whence one of the unknown quantities 

^' = ^H*^^-iHLr__....(i2) 

or sm iS cos /3 

2 TT2 

We have besides, from equation (10), u" = ~ ; whence 

' ^ ^ ' -y cos /3 

2 „ 5 r' tan /3 , 

u' = gl{ jj-j-^ -. , .... (13) 

r sm 7 

and from equation (7) 

/2 TT ^ tan/3 , .. 

u" = gll-j-. ^-i- (14) 

^ 5' sm 7 ^ ^ 

To obtain v'^ we will first make w' = u' in equation (6), which 

will give 

v''' = 2 2^'* (1 - COS 7), 

and consequently from the value of u^ (14), 

^'^ = 2 ^ H 4 *-?^^ (1 - cos 7) ... . (15). 
^ J'sm7^ ^^ ^ ^ 

Knowing u^ we obtain i^^, and, if w were required, we could 
easily obtain it by substituting in equation (5) the values of v 
and u. Thus, all the velocities may be considered as known ; 
we could deduce from them the angular velocity w with which 
the turbine must move, when it is working with the conditions 
of the maximum effective delivery ; we would then have prac- 
tically 

_u _u' 

The corresponding expense Q has for its value ^ id)' w' r' sin 7, 
or, substituting in place of w' the value of its equal u' from 

equation (14), 



TURBINES. 87 



Q = 2 -n-/ Vhh' V gB. tan /3 sin 7, ... . (16) 
a formula whose second member we should probably have to 
multiply by a number less than unity, in order to take into 
consideration the space occupied by the floats, and also to com- 
pensate for the influence of the losses of head neglected in the 
calculation. 

Let us now seek the three equations of condition to be satis- 
fied by the dimensions of the turbine. First, by dividing equa- 
tions (13) and (12), member by member, and extracting the 
square root of the quotient, we will find 
u _ h r^ sin /3 
V y r'^ sin 7' 

and by reason of eq. (8) 

sin (^ + /3) _ h r' sin /3 ^ 

sin & 5 V sin 7 ' ' * * ^ 

this is the condition obtained by eliminating eight unknown 
quantities between nine equations. There remains yet to ex- 
press that J? > 0, ^' > 0. As to this last condition, we see from 
(3) that it is itself satisfied for Fourneyron's and Fontaine's 
turbines, by supposing that they are on a level with the water 
in the tail race or below it, as we have admitted in the preced- 
ing calculations ; because then h -{- h' h positive, and we have 
2^' ^ Pa- I^ Koecklin's turbine, on the contrary, the bottom 
of the turbine is in reality above the level of the water in the 

tail race, by a positive height expressed by — (A + h^) ; — has 

for its value — , or 10°^.33 less this height ; then it is absolutely 

necessary to have 

- (A + AO < 10^.33, 
and perhaps even, on account of neglected losses of head, it 
would be well to place 



88 TURBINES. 

- (A + A') < 6". , • . . . (18) 
in order to make perfectly sure of the continuity of the liquid 
column in the cylindrical well, above which the turbine is 
found. As to the pressure^, it will be found from eq. (1) after 
substituting in it for v^ its value in eq. (19), that is 

£ = H + A + i^»-lH^;^ _^2_ (19) 

n n 2 5 />' sm /3 cos /3 ^ ^ 

The second member of this equation should, of course, be 
greater than zero ; but we may assign it a higher limit. In 
fact, if we examine the arrangement of the different systems of 
turbines, we see that there is always an indirect communication 
between the distributing orifices, situated at the end of the 
directing partitions, either with the tail race, or with the exter- 
nal air. This commimication is effected by the play necessarily 
left between the turbine proper and the distributing orifices. 
When it takes place with the tail race, p cannot differ much 
from the hydrostatic pressure ^^ + n A, which Avould take place 
in a piezometric column communicating with this race, and at 
the height of the point of entrance of the water above the 
turbine ; otherwise there would be, on account of the play that 
we have just spoken of, either a sudden gushing out, or suction 
of the water, which w^ould produce a disturbance in the motion. 
When it is with the atmosphere,^ must, for a similar reason, 
be equal to^^- It is then prudent, in the first case, to impose 
the condition that the two terms in w^hich the factor H appears, 
eq. (19), should nearly cancel each other, or, designating by h 
a number that differs little from unity, to make 

r sm /? cos /? 
Ic is moreover rigorously subjected to the condition that 

A + ^ + H (1 - /^) 



TUEBINES. 89 

should be positive. And in like manner, for the second case, 
we should establish the condition 

H (1 - K K-^Z—n) + A = A" (20 bis) 

^ r^ 2 sm /^ cos ^/ 

h^^ being a very small height. 

Again it might be proposed, for a turbine known to be work- 
ing with the maximum effective delivery, to seek this delivery 
as well as the dynamic effect. As we are supposing that all 
losses of head, other than that due to the velocity of exit v\ 
may be disregarded, the head that is turned to account will be 

and consequently the productive force (^ will be expressed by 



7/2 



2.^_ 



= 1- 



H 2^H' 

or replacing v'~ by its value 

. h tan ^ .^ . .^.. 

M- = 1 — 77 ~ (1 — cos 7) (21) 

sm 7 ^ ^ ^ 

The dynamic effect T^ would be obtained by finding the pro- 
duct (X n Q H of the effective delivery by the absolute power 
of the head ; then we would have, from eqs. (16) and (21) 



7 



T, r= n II V ^ H. 2^^ r' Vhh'. V tan jS sin 

j , 5 tan /3 , . . ) \. . . . (22) 

X n — — -, [1 — cos 7) \ 

( /sm 7 M J 

Thus have we now solved the first of the two general prob- 
lems proposed. When we take up tlie second, which consists 
in setting up a turbine for a given head, Q and H become the 
known quantities, and we have, between the nine quantities 
i6, 7, ^, r, ^r', Z>, h\ A, h\ which define the unknown dimensions, 
only equations (16), (17), (20), or (20 lis)^ to which must be 



90 TURBINES. 

added (if a Kcecklin's turbine is in question) the inequality 
(18) ; still this inequality leaves a certain margin ; and it is the 
same with equations (20) and (20 his), because the quantities h 
and h" have not a definite value. It appears, then, that the 
problem is very indeterminate, and that we may assume almost 
all the above-mentioned dimensions arbitrarily ; however, the 
following remarks impose restrictions that it will be well to 
keep in mind. 

16. Remarks on the (angles /3, 7,^, and on the dimensions J, 
y, T, r\ A, h'. — If we only considered the expression for the 
efiective delivery eq. (21), we should be tempted to make one 
of the angles /3 or 7 equal to zero ; the theoretical effective 
delivery would then become practically equal to unity. But 
we see that the expenditure Q would reduce to zero, as well as 
the dynamic effect T^ : hence the value zero is not admissible 
for either of these angles. 

Making 7 very small, the channels formed by two consecu- 
tive floats would be very much narrowed at the point of exit 
of the water; the water would flow with difficulty through 
these narrow passages, and there would be danger of its not 
following exactly the sides of the floats, which would occasion 
eddies and losses of head. On the other hand, a large value 
for 7 would diminish, very likely, the effective delivery too 
much. Between these two points to be avoided, experience 
gives a value of 20 or 30 degrees as affbrding satisfactory 
results. 

As to the angle /3, besides the reason already given, there is 
still another for not making it zero : this second reason is that 
from equation (19) j? would be negative for /S — ^ and /3 =1 90°. 
We must not then approach too closely to zero or to 90° ; the 
limits from 30 to 50 degrees have been advised by some 
experts ; but there is none that is absolute. 



TURBINES. 91 

Let us suppose that we are about to apply equation (20) ; 
multiplying it, member by member, by equation (17) we find 
h sin (^ + /3) _ 1 
sin ^ ~ 2 cos /3' 

whence 

1 ^ _ 2 cos |S sin (^ -f /3) — sin d 



h sin ^ 

or, by developing the sin (^ + /3), 

2 cos /3 sin (^ H- j3) — sin ^ == sin ^ (2 cos"" jS — 1) + 2 sin /3 cos jS cos 4 

= sin 4 cos 2 i3 + cos ^ sin 2 /3 

= sin (2 /3 + ^) ; 
we can then write 

^^^ (^ ^ + ^) ^1-1 (23) 

sin ^ ^ ^ ^ 

We have previously seen that h should be a number very near 
unity ; it follows that sin (2 /3 + ^) should be small, and conse- 
quently that 2 /3 + ^ should differ but little from 180 degrees. 
If, for example, we took /3 about 45 degrees, ^ would be about 
a right angle. Besides, it is not well to have ^ greater than 
90° ; for, if we refer to Fig. 16, we see that if ^ be obtuse the 
floats should have a form like B^ C, presenting a considerable 
curvature; and experience shows that in a very much curved 
channel the water meets with a greater loss of head, all other 
things being equal : the liquid molecules then tend to separate 
from the convex portion, which gives rise to an eddy. We see 
also in Fig. 16 that, in taking ^ very acute, the side Y U of the 
triangle B U Y — that is, the relative velocity at the entrance, 
would tend to become more or less great, which would be hurt- 
ful, since the friction of the water on the floats would be in- 
creased. Hence ^ should be an acute angle, but at the same 
time almost a right angle : we might make it vary, for example, 
between 80 and 90 degrees. 

If it be equation (20 his) and not equation (20) that we have 



92 TTJUBINES. 

to apply, the same reasons obtain for taking & acute and nearly 
90 degrees ; but the sum 2 /3 + ^ need no longer differ much 
from 180 degrees. 

After having fixed the values of /3, /, and ^, we will find from- 

h r^ . . b 

equation (17) the ratio ^^ , whence we can obtain either -— or 

— ;, the other having been assumed. 

It is in favor of the effective delivery to have -^^ less than 

unity, as formula (21) shows ; it must not, however, be greater 
than the difference ¥ — h, and must be proportional to the 
length of the fioats, in order that the channels between two 
consecutive floats may not be emptied too rapidly, because this 
emptying would give rise to a loss of head. We can impose 

the condition that V — h should be less than — the length of 

the floats. 

r' 
As we have already said, the ratio — is often taken equal to 

unity in Fontaine's and Koecklin's turbines, but it is neces- 
sarily greater than unity in Fourneyron's turbine. If it be 
taken different from unity, we must not, except for particular 
reasons, increase the difference r' — r, or r — r\ for we should 
thus lengthen the floats and increase friction. In Fourneyron's 

r' 
turbine — varies ordinarily between 1.25 and 1.60. 

The height h\ from which the water descends into the inte- 
rior of the turbine, is always zero in Fourneyron's turbines ; in 
the other two systems it is so taken that the floats may be suf- 
ficiently, but not too long, regard being had to the difference 
V — h. As to the height A, if it be a question of a turbine of 



TUEBINES. 93 

Kcecklin's, it is fixed according to local circumstances, tlie in- 
equality expressed in (18) being considered ; if it be of one of 
Fourneyron's or Fontaine's, we so arrange matters that its 
lower plane may be on tbe same level as the waters in the tail 
race when at their minimum depth. 

Finally, M. Fourneyron recommends giving the circular sec- 
tion of the tub, in which the directing partitions of his turbines 
are placed, a surface at least equal to four times the right sec- 
tion of the distributing orifices, in order that the fluid threads 
may easily pass from the vertical to the horizontal direction, 
which they must have at their point of exit. With the nota- 
tion employed in (I^o. 15), we can write 

•r ^'^ > 4. 2 -TT 7* 5 sin jS, 
or else 

^ > 8 Z> sin /3, . . . . (24) 

the sign > not being exclusive of equality. 

Let us now show, by two examples, how we shall be enabled, 
by means of these considerations, to determine the dimensions 
of a turbine to be established. 

IT. Examples of the calculations to he made for constructing 
a turhine. — Let it first be determined to establish a Fourney- 
ron turbine with the following data : 

Height of fall, H= 6^.00. 

Yolume expended per second, Q = 1"^^50. 

The absolute power of the head is 1500 x 6^^°^ = 9000 kilo- 
gramm^tres per second, or 120 horse-power. 

Since the angle 7 is not fixed theoretically, we will take it 
(]^o. 16) equal to 25 degrees ; we will also make 1c = l (^), 
which secures that j? shall be positive (No. 15) ; finally, we will 
take ^ — 90°. Equation (23) then gives 
sin (2 i3 4- ^) = 0, 
whence 



94 TUEBINES. 

2 /3 + ^ = 180° and /3 = 45°. 
As we have satisfied equation (23), which results from the 

elimination of -r-^ ^— between formulae (17) and (20), it is 
r sin ^ 

suflScient to preserve one of these last ; we deduce from both 

^1_4=: sin 25° = 0.4226 («). 

r 

The condition of expending l^'^.SO is expressed by formula (16), 
which here becomes 



1^«.50 = 2 * r' Vhh' V6 g. 0.4226, 
or else, by reducing 

/ i/0~ = 0.04785 (a^. 

We have still to express the inequality (24), which gives 

/♦ > 8 J sin 45° orr> 6.657 h; 
we will take 

r = 6h (a'O 

We have thus only three equations involving h, ¥, r, t' ; but 
on account of their particular form we may still deduce the 
values of h and r. Extracting the square root of equation (a) 
and multiplying it member by member by («'), we make r' V h' 
disappear and find 

l)r=: 0.031107, 
a relation which, combined with ^ == 6 J, gives 

r = 0^^.432, h = 0'^.072. 
This being done, the system of the three equations (a), (a'), (a'') 
would no longer give anything but r^ W\ to avoid any inde- 
termination, we will take V arbitrarily and deduce t\ except 
that the conditions mentioned in (E'o. 16), and not thus far 
expressed, must subsequently be verified. If we take, for 
example, V = 0^.090, equation (a^) will become 
/ 1/0.090 X 0.07"2"= 0.04785, 



TITRBrN-ES. 95 

whence we obtain 

These values of ^ and / may be retained, because — = 1.37, 

and the difference 2>' — 5 == O^'.OIS is only - of 7*' — r, a quan- 

y 

tity which, on account of the obliquity of the floats to the 
exterior circumference, should be but little greater than two- 
thirds of the length of these last ; the discharge of water will 
not then be too rapid. 

The height h now alone remains to be determined : if the 
level of the tail race were constant, we should make h ^= o ; 
however, we would have to consider what is said (IS^o. 16) on 
this subject. 

The theoretical effective delivery will be obtained from 
formula (21) ; we find 

, 0.072 ^ ,.o 1- cos 25° .^^^ 

fx = 1 — —— — . tan2:4o°. -. — ---r — = 0.823. 

0.090 ^ sin 25° 

In practice, we only rely upon a net effective delivery of from 
0.70 to 0.75 at most ; this it is well to do, on account of all the 
losses of head that we have neglected, and also because it is 
very difficult to make a machine move exactly with the velo- 
city and expenditure of water corresponding to the maximum 
effective delivery. 

Finally, to obtain the velocity with which the turbine should 
revolve, formula (14) should be applied, and we would deduce 

u' 
therefrom u' = 10°^.555, since the angular velocity u =z —^ =z 

17.77, and finally the number of revolutions per minute 'N = 
?^ = 169.7. 

Again, let it be proposed as a second example to set up one 



96 TURBINES. 

of Fournejron's turbines, with a head of 2 metres, expending 
0°^^60,of water per second, which corresponds to an absolute 
work of 1200 kilogrammetres, or about a 16 horse-power. We 
will suppose that the water in the tail race only rises to the 
level of the lower plane of the turbine, so that the interval or 
plaj between the turbine and the directing partitions may 
communicate directly with the atmosphere, and that the pres- 
sure j? is sensibly equal to the atmospheric pressure. We will 
then assume equation (20 his)^ making in it h" = o, in other 
words we will place 

H (1 - ^^' ^^-^—) + h^o. ■- 

6 ^^ 2 sm /3 cos /3/ 

Following the ordinary rules, we will make r := r' i besides, it 

may be remarked that, on account of the position assigned to 

the plane of the tail race, h is equal to h' with a contrary sign. 

The above equation can then be written 

H(l-^ — ^-21_)_A^=:, (^) 

^ J 2 sm i3 cos ^ / ^ ^ 

As ^ can only differ slightly from 90 degrees, we will give it 

this value ; the equation of condition (17) then takes the form 

h r"^ tan i3 _ .. . 



b'r'^ sin 7~" ' 
and, because r — r' 

y sin 7 = 5 tan ^ (5) 

Introducing V sin / in the place of h tan /3 in the expression 
(16) of the expenditure, it becomes 

Q = 2'rr^'J^sinr|/7Er. (r) 

The equations (5, 5', ^"^ are those pertaining to the problem. 
They contain six unknown quantities, viz. : /3, 7, 5, V^ t\ h' / 
we consequently see that they are indeterminate, and that we 
can assume three of the unknown quantities or three new 
equations. The angle 7 not being determinable by theory, we 



TURBINES. 97 

will take it at first equal to 30 degrees {'No. 16) ; equation {§'') 
will become, by substituting numbers for letters, 

5/ ^/ = _M2,^ = 0.04312, 

a relation which is satisfied by the values 

r' = 0^60, y = 0^.072. 

We thus see that the ratio ^is only 0.12; consequently the 

inequality in the velocities of the liquid threads in the orifice 
of exit will not be too noticeable. Now as there remain three 
unknown quantities, h\ h\ /3, connected only by the two equa- 
tions (^), (5'), we will assume h' = 0°^.15 ; then eliminating 

-J- between {§) and (^^), we will obtain 

^ 1__^ ^ _ A 

2 cos^ /3 "" H "" 40' 
whence 

2 cos^ I3 = ~,2 cos^ /3 - 1 = cos 2 i3 ^, 
oi 37 

and consequently 

/3 = 42° 40 very nearly. 

Knowing j8, we obtain from equation ((5') 

h = 0^.039. 

The difibrence h' — h = 0™.033 is perhaps too great relatively 

to the height 0°^.15 of the turbine, because the floats having a 

development of from 0°^.20 to 0°^.25 at the most, their spread 

would reach the amount of - very nearly. We then try an- 
other value of h^ ; for example, let 

h' = 0°^.30. 
Proceeding as before, we will have successively 

QA QA 

2 cos' p z= ^, cos 2 /3 = ^, /3 = 40°, h = 0^.043. 



98 TUEBINES. 

The difference 5' — 5 would then be0"^.029; but as the floats 
would be nearly 0°^.40 in length (on account of their inclination 
to the lower plane of the turbine), this number appears per- 
fectly admissible. We should then obtain the results 
7 = 30°, /3 = 40°, & = 90°, r = r' = 0"^.60 
h =: 0^^.04:3, h' = 0°^.072, -h = h' = 0^.30. 
The effective delivery ^ would be given by equation (21), 
which, considering equation (5^), becomes 

fx = cos 7 = cos 30° = 0.866. 
In like manner equation (14) would be simplified and give 

whence 

u' = 4°^.429, 

from this we deduce finally the angular velocity to be given to 
the machine 

c^ = X = 7.382, 
r 

and the number of revolutions per minute 

N =. 1^-^ = 70.5. 

In general, as we have seen, the problem which consists in 
fixing the dimensions of a turbine for which we have the ex- 
penditure and the head is indeterminate; we take advantage 
of this to assume in part the unknown dimensions, and by the 
method of trial, if that be necessary, endeavor to satisfy the 
different conditions to be fulfilled, but which the equations do 
not express. 

18. Of the means of regulating the expenditure of water in 
turbines. — A turbine constructed with assigned dimensions, in 
order to move with the maximum effective delivery, should ex- 
pend a perfectly determinate volume of water, so long at least 
as the head remains constant. However, in practice, we are 



TURBINES. 



obliged to regulate the expenditure by the volume furnished 
by the pond ; for if we expended more we would be exposed to 
the lack of water, after some time, and we would then be 
forced to suspend the operation of the machine. Consequently 
we so calculate the dimensions as to expend suitably the great- 
est volume of water at our disposal, and we make proper dis- 
positions to expend less when the supply diminishes. For this 
purpose several methods have been employed. 

In Fourneyron's turbine, the movable tub E G F I (Fig. 13) 
allows this end to be attained : it is merely necessary to lower 
it more or less, to contract the openings G K, I L, or to close 
them entirely. The vertical motion of translation of this 
tub is obtained by means of three vertical rods, as r 5, t ii, 
which are attached to it at three points, which form the vertices 
of an equilateral horizontal triangle. These rods are termina- 
ted at their upper ends by screws, and enter into nuts which are 
forced by their construction to turn in their places. The three 
nuts, moreover, are furnished with three cogged wheels, all 
exactly alike, which are geared on the same wheel, which is 
loose on the axle of the turbine. By turning one of the nuts 
by means of a wrench, the other two are turned exactly the 
same amount, and the tub is raised or lowered by the three 
rods at once. 

There is one great inconvenience attendant upon the partial 
obstruction of the openings G K, I L ; it is that the fluid veins 
which issue through these openings immediately enter canals 
of greater section, in which they flow necessarily through a full 
pipe, since the turbine is below the tail race. There is a sud- 
den change of section thus produced, and consequently a great- 
er or less loss of head. This influence is sometimes so great 
that General Morin has mentioned, in difl'erent experiments on 
a turbiiie, a diminution in the eifective delivery from 0.79 tO' 



100 TURBINES. 

0.24, when the free opening nnder the wheel descended from 
its greatest elevation to about -J- of this height. The inconve- 
nience is so mnch tlie greater as the diminution in the effective 
delivery corresponds to tliat of the volume of water expended, 
Avhich tends to make the dynamic effect of tlie machine very 
irregular. To remedy this, M. Fourneyron has proposed to 
subdivide the lieight of the turbine into several stages, by means 
of two or three annular horizontal plates, like the plates 
S R M N, U T P Q, the distance between which is divided into 
three or four equal parts. Supposing, for example, that there 
are three stages, we see that there will be no sudden change of 
section when the cylindrical sluice is raised |-, f , or the whole 
height of the turbine ; in all cases the phenomeuon of sudden 
expansion will only affect a portion of the liquid vein. But, 
on the other hand, the construction of the machinery is compli- 
cated, and the friction of the water against the solid w^alls is 
increased. M, Fourneyron has again proposed using only the 
two plates S R M N, U T P Q ; the lower one would carry the 
floats only, and the upper one be pierced with grooves which 
would allow it to settle freely between the floats, under the 
action of its weight alone. This upper plate would bear on a 
flange placed at the bottom of the cylindrical sluice on the out- 
side. When the sluice is lowered, the plate S R M E" is low- 
ered with it, and the height of the turbine would be always 
equal to the height to which the sluice is raised. The movable 
plate carried around by the turbine would, while turning, rub 
against the flange which serves it as a support ; but, the pres- 
sure between the two being slight, this would not produce an 
increase of resistance worthy of mention. 

M. Fontaine, to regulate the expenditure of his turbines, 
uses a series of valves similar to that represented in Fig. 17. 
A B is a directing partition, B C a float of the turbine, D a 



TURBESrES. 



101 



valve which can be sunk to a great or less depth into the space 
between A B and the next directing partition to the left. In 
this way we can narrow as much as we choose the free pas- 
sage into this interval, and as the same effect is produced on all 
in the same way, it is plain that we have the means of reducing 
the volume of water expended as much as circumstances may 
require. The vertical motion of translation is given simultane- 
ously to all the rods E F by means sim.ilar to those used by 
M. Fourneyron : these rods are all 
connected with a metallic ring, 
at three points of which are at- 
tached vertical screws, furnished 
with nuts which can be turned 
only in their beds. These three 
have each a cogged wlieel secure- 
ly attached, the wheels being all 
of the same size, surrounded by 
an endless chain, which causes 
them to turn equally and at the 
same time. It is then necessary to turn one of the three 
wheels by means of a winch and pinion, in order that the whole 
system of valves may take up a vertical motion. The partial 
closino; of the in-leading; canals here, as in Fournevron's tur- 
bine, is not without inconveniences, for the sudden change of 
section in these canals still gives rise to a loss of head ; how- 
ever, this loss is found to be diminished to a considerable 
extent. 

For the valves M. Koecklin has substituted clack-valves re- 
volving on a hinge, so as to fit exactly over the entrance to the 
distributing canals, when the closing is complete. The ar- 
rangement of the Koecklin turbine also allows the expenditure 
of water to be regulated by making use of the sluice Y (Fig. 




Fig. 17. 



102 TUKBmES. 

15), which can close the communication between the well 
placed below the turbine and the tail race. But experience 
shows that a greater portion of the head is lost in this way than 
by using the valves. 

The inconveniences attendant upon the partial closing of the 
distributing channels are so great, when considered from the 
point of view of economy of the motive power, that all possible 
means for remedying them have been sought for. We have 
already mentioned two devised by M. Fourneyron. M. Charles 
Gallon, civil engineer, and a constructor of reputation, has pro- 
posed another way, which consists in making all the partial 
sluices, which close the channels in question, independent of 
each other ; to diminish the expenditure, a certain number of 
these sluices can be closed completely, leaving the remainder 
entirely opened. But as the channels formed by the floats of 
the turbine pass alternately before the opened and closed ori- 
fices, there is still a cause of unsteadiness and trouble in the 
motion. 

JVI. Gallon's idea has been reproduced under another form by 
M. Fontaine. The orifices of entrance of the distributing chan- 
nels occupy a horizontal surface comprised between two circles 
concentric with the axis of the turbine, M. Fontaine arranges 
two rollers, of the shape of a truncated cone, which can roll 
over this annular surface. The two rollers are mounted on the 
same horizontal spindle which forms a collar surrounding the 
axle of rotation. When they move in one direction, each one 
of them unrolls a band of leather, which has one end fastened 
to the roller and the other to the plane of the orifices of en- 
trance : some of the orifices are thus entirely closed while the 
others remain wide open. When the truncated cones move in 
the contrary direction, they roll up the two leathern bands and 
uncover the orifices. M. Fontaine has also imitated the tur- 



TURBINES. 103 

bines of several stories of M, Fourneyron, in proposing to di- 
vide the turbines into several zones by means of surfaces of 
revolution about the axis of the system ; each of these zones 
could be separately obstructed. 

19. Hydropneumatic turbine of Girard and Callon, — The 
problem of regulating the expenditure, without too great loss, 
appears to have been solved in the best manner in a kind of 
turbine called by its inventors, MM. Girard and Callon, the 
hydropneumatio turbine. Their system consists essentially in 
surrounding Fourneyron 's turbine with a sheet-iron bell, the 
lower plane of which is nearly at the height of the points at 
which the water leaves the floats. In this bell, by means of a 
small pump set in motion by the machine itself, the air is com- 
pressed, which gradually forces the water entirely out of the 
bell ; then, if we suppose the cylindrical sluice to be partially 
raised, the liquid vein which escapes below has a depth less 
than the distance between the two plates of the turbine ; but 
no sudden change in the section of liquid results from this, be- 
cause the turbine moves in compressed air, and it is not cover- 
ed by the water in the tail race. The water flowing into the 
turbine has a depth which, at the beginning of the floats, is 
equal to the height to which the sluice is raised ; the upper 
plate is no longer wet, and as it only serves to hold the floats, 
it can be hollowed out, in order to make sure of a free circula- 
tion of air above the liquid vein. Thus the principal cause of 
the loss of head, due to the partial raising of the sluice, is 
found to be suppressed, and we should succeed in obtaining a 
very slightly varying eflective delivery. 

The calculations to be employed for the hydropneumatic tur- 
bine may be regarded as a particular case of those in l^o. 15. 
Preserving the same notation, we must consider V as an un- 
known quantity, and at the same time suppose 



104 TURBINES. 

in fact, A represents the depth to which the turbine is immersed 
below the level of the tail race, and ^^ + n A is properly the 
pressure of the air in the bell. Thus, from this value oip equa- 
tions (19) and (20) show first that ^ == 1, and consequently (No. 
16) that we have 2 iS + ^ = 180°, the only condition to be satis- 
fied by the dimensions of the machine. It gives /s -|- ^ = 180° 
— /3 ; equation (IT) then becomes 

h' 7''^ sin 7 = Z> r^ sin ^ == J r^ sin 2 jS ; 
whence we can deduce h' in terms of the height h to which the 
sluice is raised, for a turbine moving with the maximum efiec- 
tive delivery. In virtue of tlie preceding relation, equations 
(13), (16), and (21) take the form 

^ sin/ 2cos'/3' 



Q =: 2 ^ J r sin /3 y 2 ^ H, 

these formulas, which can very readily be demonstrated direct- 
ly, give : 1st, the angular velocity - of the turbine, which an- 
swers to the maximum effective delivery; 2d, its expenditure 
Q, and its effective delivery m- under the same condition. If we 
had to set up a turbine for a given head and expenditure, the 
equation of which Q is the first member, together with the in- 
equality (24) (No. 16), would give us the means for finding ^' 
and h sin /3, and consequently &, ^fter j3 has been chosen. As 
to 7, it should always be taken between 20 and 30 degrees ; & 
should be equal to 180° — 2 jS; finally, r' should be taken as 
small as possible (on account of the expression for fj-), but not, 
however, so as to run any risk of making the floats too short. 

The method of MM. Girard and Gallon could be equally well 
adapted to Fontaine's turbine. 



TuiiBmES. 105 

, 20. Some practieal views on the subject of turhines. — The 
directing partitions and floats are generally made of sheet iron ; 
thej are fastened to the surfaces that are to support them either 
bj angle irons, or by setting them in cast-iron grooves on these 
surfaces. They should be sufficient in number to give to the 
velocity of the water their own direction. The distance of any 
two consecutive floats or partitions apart should not be, at any 
point, more than 0°^.06 to 0°^.08, measured along the normal to 
the surfaces, and generally it is made less. However, it must 
not be made too small, for then the friction of the water against 
the solid sides would be too great. 

As the floats in Fourneyron's turbine are placed further from 
the axis than the partitions, that is to say distributed over a 
greater circumference, their number is from one-third to one- 
half greater than that of the partitions, in order to have every- 
where a suitable distance apart. 

Excepting the condition of cutting the planes of the orifices 
under a determinate angle, the curvature of the floats and par- 
titions is almost a matter of indifi'erence. However, as too 
great a curvature, or a sudden change in the curve, may pre- 
vent the threads from following the sides, and thus produce 
losses of head, we must avoid these two faults. It would be 
well to have the radius of curvature at least three or four times 
the distance apart measured along the normal. 

Turbines can be used for all heads and every expenditure. 
For example, some are mentioned the head being only from 0°^.30 
to 0°^.40, whilst, in the Black Forest, there is a turbine set up by 
M. Fourneyron which lias a fall of 108 metres. The expendi- 
ture may be great even with quite small dimensions. In one 
of the examples given in InTo. 17, we have seen that a turbine 
of 0°^.60 external radius and 0°^.09 in height, expended without 
loss 1°^°.50, or 1500 litres per second. Some turbines are con- 



106 TUKBINES. 

structed whose expenditure reaches as high as 4 cubic metres, 
or 4000 litres, per second, and, if need be, more could be ex- 
pended. 

Under ordinary circumstances, turbines move with sufficient 
rapidity, and thus allow the gearing for transmission to be 
economized. 

For each turbine organized with fixed conditions, the theory 
of ISTo. 15 gives a certain velocity to be imparted to it, in order 
to obtain the maximum useful effect. But if, in reality, a velo- 
city different from this be given to it, and deviate from 25 per 
cent, more or less, experience shows that the effective delivery 
does not change much, a very important property for many 
manufacturing purposes, in which, in spite of the variations 
that take place in the expenditure of the head of water, it is 
important to have the machine always move with a very nearly 
constant velocity. 

These motors have then a very decided advantage over 
wheels with a horizontal axis. Unfortunately their propor- 
tional effective delivery is not always constant, even approxi- 
mately, for heads with a very variable expenditure ; their con- 
struction and repairs can only be intrusted to very skilful me- 
chanics, and consequently are quite expensive ; while, on the 
other hand, overshot and breast wheels can be very economi- 
cally constructed, and at the same time give an effective delivery 
at least equal to, if not greater than, that of turbines. These 
wheels will frequently, on this account, be preferred when the 
expenditure and head of water are favorable to their con- 
struction. 



REACTION WHEELS. 



21. Reaction wheels. — Let us conceive of one of Fourney- 
ron's turbines deprived of its directing partitions, care being 
taken to prolong the floats to a slight distance from the axis of 
rotation ; let us suppose that the water comes to the floats 
through a pipe concentric with the axle, having for a radius 
precisely the free distance that we have just mentioned. Fur- 
thermore, the expenditure of water in this pipe will be consid- 
ered as very little, in order that the absolute velocity of the 
liquid in it may not be sensible. We shall thus have the idea 
of reaction wheels. 

To give their theory, we will consider the point of entrance 
of water into the wheel as being on the axis of rotation itself; 
at this point the velocity of the wheel being nothing, as well 
as the absolute velocity of the water, it will be the same for 
the relative velocity of this last. In the calculations of ISTo. 
15, we will have to make v ^=^ o, u =^ o^vj ^:^ o ; equations (1), 
(2), and (3) then become 

o = ^ + h+ £±I1^ 
n 



w 



whence we find 



S^ + h + h'^Pi 
n n 



w" = 2gB.-\- u'\ 



108 KEACTION WHEELS. 

JS'eglecting friction, the only loss of head is as yet the height 

— due to the absolute velocity of exit ; we calculate its value 
«/ 
by means of equation (6), which, combined with the preceding, 

will give 



?;'' = 2 ^'^ + 2 ^ H - 2 ^^' cos 7 y 2 ^ H + u'\ 
The effective delivery would then be 



0)'^ 

H-^""- 



2 a , v" u'' u' 



v^^.i;. 



, . u' 
or, placmg — = x 

VgR 

IX z= — x^ -\- X COS 7 y 2 + cc^ 
"We can consider /x as a function of x, and seek its maximum 
when X varies. To this end we will take off the radical sign 
by writing 

(^ + xj = x' cos' 7 (2 + x'), 
or by reducing 

x" sin' 7 — 2 a?' (cos' 7 — fx) + H-'' = 0. 
If we deduced from this a?' in terms of fx, the roots would 
become, from their nature, real; consequently, we have the 
condition 

(cos' y — i^y — ijJ^ sin' 7 > 0, 

or developing and reducing 

cos" 7 — 2 (X cos' 7 + fx' cos' 7 > 0. 
If we suppress the positive factor cos' 7, we find 

cos' 7 — 2?^ + fJ-' > 0, 
or 

(l-f^y-sin'7>0, 



EEACTIOX WHEELS. 



109 



and, observing tliat 1 — ,u. is necc-sari^j pj3itl\e, ai wcJ a^ 
sin 7, 

1 — /x > sin 7, 
fjt- < 1 — sin /. 
The limit of the effective delivery that we can reach is then 

1x^ = 1 — sin y. 
The corresponding value for x is easil}^ obtained bj the relation 
between x and m- ; if we take the equation deprived of radicals, 
and make in it a? = x^ and f;. = 1 — sin y, it becomes 

a?,* sin' y — 2 x^"" sin / (1 — sin 7) + (1 — sin 7)' = 0, 
or, more simply, by extracting the square root 



sin 7 — (1 — sin 7) = 0, 



whence 



_ / l-sin7 

'^-y sin 7 • 



If we supposed y = 0, we would find ij.^ = 1 ; but a?j, and 
consequently v/, would become infinite. Strictly speaking, the 
value y = might be realized ; it would only be necessary 
that the channels that make 
up the wheel should be ar- 
ranged no longer side by 
side without empty spaces, 
as in turbines, but according 
to the annexed sketch (Fig. 
18). We should have a 
certain number of curved 
plates, as A B, meeting the 
circumference OB at B, 
where they end. and com- Fig. is. 

muuicating at A with the supply-pipe, to which they are per- 
manently attached. The supply-pipe would then form the 
axle of rotation. But, in adopting this arrangement, v/ could 




^ iV 



110 EEACTION WHEELS. 

not become infinite, nor fjt,, consequently, reach unity : we only 
see that it would be necessary to make the wheel turn very 
rapidly. Moreover, it is difficult to expend a large volume of 
water without giving a considerable diameter to the central 
pipe A O, which would make the hypothesis, that u^ v, and w 
were equal to zero, inadmissible ; there are equally great diffi- 
culties in regulating the expense according to the required cir- 
cumstances. It is undoubtedly on these accounts that this kind 
of wheel is but little used. 

Preserving the arrangement of the floats of Fourneyron's 
turbine, which gives a series of contiguous channels, we can no 
longer make 7 = 0, and then the limit of the theoretical effec- 
tive delivery decreases quite rapidly as / increases ; so that, for 
7 = 15°, 1 — sin 7 would no longer be greater than 0.741. On 
the other hand, as we ofier the water a wider outlet, we should 
perhaps lose less in friction, and the theoretical effective deliv- 
ery would differ less from the real. 



PUMPS. 



§ IV. Of A FEW Machines eok Raising Water. 

22. Pur)ijps. — The arrangement and shape of the parts of 
pumps are of infinite variety, according to the notions of the 
constructor. A special treatise would be necessary merely to 
describe the principal kinds. We will suppose, then, that the 
reader has seen a summary description of these machines, and 
confine ourselves to general ideas. 

{a^ Effort necessary to make the piston move. — Two cases 
must be distinguished : pumps of single stroke, and pumps of 
double stroke. In the former case, the piston only draws up 
the water into the pump, or else only drives out the water pre- 
viously drawn up, by forcing it up through the delivery yjipe, 
when it moves in a determined direction ; in the second, these 
efi'ects are produced simultaneously, whatever be the direction 
of the stroke. Let us first take the single stroke ; let 

h be its height ; 

n the section of the piston ; 

jPa the atmospheric pressure ; 

n the weight of a cubic metre of water. 
The side of the piston in contact with the column of water drawn 
up would support, supposing that it remained in equilibrio, a 
pressure equal to n {pa — n A), whilst the other side, generally in 
direct communication with the atmosphere, would sustain a 
pressure in a contrary direction equal to j?^ ^ ; the difterence 



112 PTJMPS. 

nnh would be the resultant pressure on the piston. If, on 
the contrary, there be but a single stroke forcing to a height h\ 
we will find, in like manner, that the piston supports, exclu- 
sively of its motion, a resultant pressure n n h! , Finally, if 
the pump were one of double stroke, these two resultants would 
be superposed, and the value for the total pressure w^ould be 
n £2 (A + A') or n n H, H being the height included between 
the level of the basin from which the water is drawn, and that 
of the basin receiving it. In a certain class of single-stroke 
pumps, the same superposition of resultant pressure on the two 
faces of the piston ]:akes place during the motion in one direc- 
tion, and these resultants are in equilibrio when motion occurs 
in the opposite direction : this is the case in the kind of pump 
called the lifting-jpum.p. If the piston have no horizontal 
motion, we must, besides, consider its weight as well as that of 
its rod, the component of which, parallel to the axis of the 
pump, would be added to or subtracted from the preceding; 
expressions, according to circumstances. The friction of the 
piston against the barrel of the pump must also be added, as 
well as that of the rod against the packing-box, if there be any, 
through which it passes. 

But these expressions only give the value of the force capable 
of maintaining the piston, as well as the water drawn or forced 
up, in equilibrio in a given position. When motion takes 
place, the effort brought to bear on the piston may be very dif- 
ferent from this force. In the first place, the water does not 
begin to move in the pipes, and does not pass through the nar- 
row openings of the valves without experiencing losses of head 
which are to be added to the heights A and h! . For ex- 
ample, in the case of the sucking pump, if there be a loss of 
head ? in the length of the column drawn up, the pressure 
n (^^ — n A) will be reduced to n (^^ — n (A -f- ^ ), and the 



PTJMPS. 113 

resultant n n A would become n n (A + ^). In like manner, 
if we consider a single forcing stroke, and that there is in the 
entire column forced up a loss of head ^', the piezometric level 
in this column, at the point at which it touches the piston, 
would be increased by l\ and the resultant pressure would be 
n n (A' -f- ^'), If the pump be one of double stroke, the ex- 
pression n n H should in like manner be replaced by n a (H 
+ ? + D- Besides the heights I and l\ others must be added, 
if the mass of water set in motion is not uniformly displaced. 
LetP be the weight of the piston and its rod,y its acceleration, 
P' the weight of the water set in motion and which fills the 
pipes through which it is drawn up or forced out : the weight 
P^ being supposed to move with a mean acceleration j\ we see 

that an additional force - (P^ + ^' j') would be necessary to 

overcome the inertia of the water and piston. This additional 
force, at one time a pressure, at another a resistance, may pro- 
duce considerable variations in the total force to be applied to 
the piston, which is always inconvenient : since we must, in 
the first place, determine the dimensions of the pieces, not ac- 
cording to the mean, but according to the maximum efibrt sus- 
tained, which generally produces a clumsy and expensive 
machine ; in the second, it is seldom that great variations in 
resistance do not give rise indirectly to some loss of motive 

power. We diminish the value of the term — - by means of 

weights acting as a counterpoise to the piston, if P is large ; f 

and consequently — — are also diminished, in case it is found 

to be worth the trouble, by means of an air-chamber, placed 
at the entrance to the ascent pipe, which makes the motion of 
a great part of the weight of the column forced up uniform. 



114 



PIJMPS. 



and consequently suppresses or diminishes to a very great 
extent the corresponding force of inertia. 

Another means of obtaining approximate uniformity of mo- 
tion in the ascent pipe consists in making it answer for the 
delivery of several pumps working together, in such a way 
that their total delivery in a series of equal 
times shall vary but little : it is accomplished 
in this way. Let O (Fig. 19) be the axis of 
rotation of an arbor that receives from a 
motor a motion rendered regular by a fly- 
wheel, and consequently nearly uniform. 
This arbor carries two arms, O B, O B', mak- 
ing a right angle witli each other ; to each 
of them is attached a connecting rod, fasten- 
ed at its other end to a piston running be- 
tween guides, and which belongs to a double-stroke pump. As 
an example, we will suppose the arbor O horizontal, the piston 
rods vertical and their prolongations intersecting the axis of 
rotation ; the connecting rods will generally be of the same 
length, about five or six times as long as the arm O B. It fol- 
lows from this that the obliquity of the connecting rods with 
the vertical being always small, the velocities v and v' of the 
pistons are practically those of the projections of B and B' on 
the vertical B„ B, ; calling « the angular velocity of the arbor, 
h the length O B, a? the angle formed by O B with the vertical 
B„ B,, we will then have 




Fig. 19. 



'y = w 5 sin cc, v' 



J sin 



m(^ + a;) 



h cos 



X. 



Let n again be the common cross-section of the two pistons, 
and 5 a very short interval of time ; exclusively of the losses 
by the play of the two mechanisms, tlie volume of water fur- 
nished during the time ^ to a common ascent pipe, by the two 



PUMPS. 115 

pumps together, will be the arithmetical sum of the volumes 
generated by the two pistons, viz., n (v -^ v')& or n u h& (sin x 
4- cos x), a formula in which the sine and cosine should have 
their absolute values given, since it is a question of an arith- 
metical sum, and therefore the velocities are essentially positive. 
Consequently it is sufficient, in order to obtain the maximum, 
the minimum, and the mean of the variable quantity sin x + 

tjS' 

cos X, to suppose x included between o and -. l^ow we find be- 
tween these limits 

Two minima equal to 1 for a? = <9 and a? = ^ 

One maximum equal to 1.414, for x= j 



It 



f2 

The mean value equal to — ^^ — = - = 1.272. 



r 



dx 

There would thus be between the minimum and the mean a 

272 
relative difference of t^--, or about 0.214 : whilst, with a sin- 
1.272' ' ' 

gle pump, the elementary delivery, proportional to sin cc, would 

2 
vary between and 1, and would have - or 0.637 for a mean 

value, which would produce a much greater relative difference 
between the minimum and the mean. 

We obtain a still more satisfactory result when we use three 
arms making angles of 120° with each other. The elementary 
delivery of the three pumps together is then proportional to 

/ 2 "^ \ . / 4 'TT'X 

SHI ^' + sm f £c + -— j + sm f a? 4- — J, 

each sine to be always taken as positive, whatever may be x,. 
8 



116 PUMPS. 

We readily see, moreover, that the arithmetical sum of the 

three sines will not change by increasing the arc by 60 degrees, 

tff 
so that it is sufficient to make x vary between and -. Within 

these limits, the first two sines are positive and the third nega- 
tive ; hence, the sum of the absolute values will be expressed 

by 

sm X + sin (x + — j — sin (x +— \ 

or else, by developing and observing that the arcs — and — 

o o 

together make an entire circumference, 

sm a? + 2 cos x sm --- 

or finally 

sin a? 4- V 3 cos x. 

The minima of this quantity correspond to a? = and x ^ 
and have for their value V 3 or 1.732 ; the maximum, corre- 
sponding to a? = -, is - + 7^ '^^ 3. V 3, that is 2 ; the mean 
6 2 2 

f^ . /- 

J „ (sin X -\- y Z GO^ X) d X ^ 6 -, A^ a rni 1 . • 

— ^-^ ^ becomes - or 1 .910. The relative 

rs 

*/„ dx 
difference between the minimum and the mean is consequently 

-[ 9IQ 2 73-^ 

lowered to — — :r-^^ — - or to about 0.093. 

There is also a great deal of regularity in the elementary 
delivery of the three pumps united as above, w^hen they are 
supposed to be of a single stroke. Let us admit, for example. 



PTJMPS. 117 

that each piston only forces up water when its crank O B de- 
scends from Bo to B^ ; the snm of the elementary deliveries will 
stil] be proportional to the expression 

sm X + sinf £c -f — J + sm [x + -^\ ; 

but the pumps being only of single stroke, instead of changing 
the sign of the negative sines, we must suppress them altogether. 

This granted, let us first increase x from to ^ : x and x 4- 

o 

— - will be less than the semi-circumference, and a? + -— will be 
o 3 

included between -jt and 2 'k. We shall then only have to keep 

a? -h — j, which can 
be put under the form 

2 sin [x ^^ cos -, or sin [x + \ 

since cos -=: --^ this sum, equal to sin o, or 0.866 for a? = 0, 

If 
becomes a maximum and equal to 1 for a? = ^, then it decreases 

to 0.866 when x passes from ^ to -. In the second place, if we 

u o 

take the values of x between - and — , the sines of a? + — - and 

o 3 3 

a? + — are both negative, so that the sin x alone can be pre- 
served, which has still a minimum value 0.866, corresponding 
to two limits, and 1 for a maximum equidistant from these 
limits. Moreover, it is useless to consider the values of x 

2-^ 
greater than — , for a rotation of 120 degrees not producing 
o 

any change of figure in the combination of the mechanism, we 



118 PUMPS. 

would again find the same sines. We see, then, that the ele- 
mentary delivery of the three pumps working together varies 
as certain numbers which are always comprised between 0.860 
and 1, and consequently that it is sufficiently regular : the 
minimum and maximum are respectively half of what they 
were in the case of three pumps of double stroke. 

We have heretofore supposed that the two cranks at right 
angles to each other, or the three cranks following each other 
at distances of 120 degrees, are fixed to the same arbor ; it is 
plain that they can be attached to different arbors, provided 
that they all have the same angular velocity, the cranks being 
of equal length ; or, more generally, provided that the centre 
of the joint of each of them with the corresponding connecting 
rod, in the three systems, has the same velocity of rotation 
about its arbor. 

It is always well, as has been already said, to avoid great 
variations in the force that is to be transmitted to the piston of 
a pump ; this becomes almost indispensable when it is moved 
by means of horses. An essential condition for employing the 
work of horses to advantage is that their pace and the exertion 
they have to make shall be as steady as possible ; it would be 
difficult to accomplish this with one single stroke pump forcing 
up a long column of water, the piston of which should receive 
its motion from the arbor of the gearing-wheel by a system of 
a crank and connecting rod, and we could only succeed by em- 
ploying fly-wheels, more or less clumsy. It would generally 
be better to regulate the resistance by means now to be men- 
tioned. 

(5) WorJc to he transmitted to the piston. — If we knew exact- 
ly, in each position of the piston, the force to be applied to give 
it motion, it would be easy to deduce the amount of work to 
be transmitted to it. But this force cannot be exactly deter- 



PUMPS. 119 

mined ; tlius, the determination of friction of the piston against 
the pump barrel, or against the packing that it passes through 
(if it be a plunger), is necessarily uncertain, because it depends 
on the skill of the constructor ; we can say the same for the 
losses of head encountered in the sucking and forcing pipes, on 
account of the defect of permanence and uniformity in the 
motion. However, when the forcing pipe is very long, we 
have seen that it is well so to arrange matters that the motion 
in it shall be uniform, and then we can calculate with sufficient 
exactness the entire head ^'' between the two extremities of 
this pipe. This granted, let us admit first, that we are consid- 
ering a double-stroke pump : the resultant pressure exerted on 
the piston being expressed by n n (H + ? + T)? the force to 
be transmitted to it will be represented by n ^ (H + ^^') + F. 
We will include in the additive term F the friction against the 
barrel of the pump and the packing, the excess of ^ + ^"^ over 
^^\ and lastly inertia. The total work of this force, in a linear 
distance I, divided into elements d a?, will be 



nnZ(H + n -^f^Fdx. 



Moreover, n I represents very nearly the volume of water raised 
by one stroke of the piston ; if then we wish the work expend- 
ed in raising each cubic yard of water to the height H, we will 
have to calculate the quantity 

In practice, on account of the difficulty of determining exactly 
the integral^ Fdx^ we simply multiply the term n (H + Z,") 



by a co-efficient n such as 1.10 or 1.15 or 1.20, according to 
the greater or less perfectness of the machine. 

If the pump were one of single stroke, we would obtain the 



120 PUMPS. 

same expression for the work, by adding together two consecu- 
tive strokes of the piston. 

When we wish to determine the amount of horse-power to 
be transmitted to the piston, we must also know the mean velo- 
city u of the piston. We easily deduce the mean amount 
pumped up per second Q u^ if the pump is one of double stroke, 

or — -— for a single-stroke pump ; we multiply this amount by 

71 n (H 4- J^') ; dividing finally by 75, we have the horse-power 
sought. 

(c) Mean velocity of the piston ; delivery of pumps. — The 
mean velocity of the piston ought not to be very small, because, 
in order to pump up any considerable quantity of water, we 
would have to make the body of the pump and the lifting pipe 
very large, which would increase the cost of construction. But 
too great a velocity possesses also great inconveniences : firstly, 
we increase the losses of head in a very rapid proportion ; then 
it may happen that the water furnished by the sucking pipe 
may not come up sufficiently fast to follow the piston, and that 
the pump barrel may not be filled at each stroke, which would 
cause a loss in the delivery, and a shock on the return of the 
piston in an opposite direction. On account of the difficulty 
of determining exactly the velocity of the water sucked up, we 
ordinarily adopt a mean velocity of the piston, about 0™.20 per 
second ; we rarely go so high as 0°^.30. It is clear that the 
limit may be increased in proportion as the piston moves to a 
less height above the basin from which the water is drawn, and 
the more care that has been taken to avoid losses of head in 
the sucking pipe. 

The piston having a stroke the length of which is I and the 
cross section n, describes, during one of the periods employed 
in raising water to the upper basin, a volume ^l ; this volume 



PUMPS. 121 

would also be that of the water raised dui'ing the same time, 
if there were no leakage around the valves between the piston 
and the hollow cylinder in which it moves. On this account, 
the volume raised varies from 0.75 n Z to n Z/ the co-efficient 
by which the volume described b}^ the piston is to be affected 
varies with the care displayed in the construction and in keep- 
ing the pump in good order; under ordinary circumstances, we 
may take it from 0.90 to 0.92. 



SPIRAL NORIA 



23. Sjpiral Noria. — This wheel consists essentially of a hori- 
zontal arbor O (Fig. 20), to which are fastened a certain num- 




Fig. 20. 



ber of cylindrical surfaces having their generatrices parallel to 
the axis ; the right sections of these cylinders are involutes of 
a circle. The space between two consecutive cylinders thus 
forms a canal with a constant breadth, as well in the direction 
of the normal as perpendicularly to the plane of the figure. 
One of the canals, for example, will have its outer opening at 
A B and the other at I G. The entire system turns around 
the axis O, in the direction of the arrow-head ; the centre O is 
above the basin whence the water is drawn, and the level of 
this basin covers the lower part of the wheel more or less. 
During the time that the opening A B is wholly or partially 



I 



SPIRAI. NOEIA. 123 

below the surface level of the basin, a certain quantity of water 
enters the canal A B I G, bj the effect of the rotation ; the ro- 
tation continuing, A B rises until finally it comes directly over 

1 G ; then the water taken in flows into I Gr through holes left 
open all around the arbor, and falls into a canal which leads it 
off to the basin that is to receive it. 

We will propose two questions : 1st, a given spiral noria 
turns with a known angular velocity, and occupies a given po- 
sition with respect to the lower basin; what will be the amount 
raised per second ? 2d, what will be the work that the motor 
will have to transmit to it ? 

Let us call S the section A B projected on the plane passing 
through the axis O and the centre of A B ; N the number of 
revolutions of the wheel per minute ; n the number of invo- 
lutes ; H the height O C of the point O above the level of the 
water to be raised; r\ t" the distances of the points A and B 
from the axis of rotation. The point A will describe under- 
neath the water an arc DAD', of which we will designate the 
angle at the centre by 2 a ; in like manner the point B will de- 
scribe the arc E B E', corresponding to the angle at the centre 

2 /3. First we shall have 



cos a = — COS /3 = — -, 



-1 



H 



arc D A D' = 2 T'' a :r= 2 T'' cos =i, 



arc E B E' = 2 r^' /3 = 2 r" cos 






the arc described below the water by the centre of A B differ- 
ing little fi 
pressed by 



ing little from the mean - (D A D' -f- E B E') will then be ex 



r cos -J 4- r cos -^^ = L. 



124 SPIRAL NORIA. 

Now, the volume that entered at the opening A B is E D E' J)\ 
or the product of this arc by the perpendicular section S ; then, 
since there are n canals that raise the same volume in each 
revolution of the wheel, the volume raised will be, per revolu- 
tion, ?^ L S ; finally, the number of revolutions per second being 

— , the amount Q raised by the wheel in the same time will be 

expressed by — N ti L S, that is, we ought to have 

1 ~^H ~^II\ 

Q = -— 1^ n S (r^ cos -7- + r" cos -7. j. 
60 r r y 

But this calculation supposes that there enters, during each 
element of time d t through the opening A B, a volume of 
water equal to that generated by A B in the same time ; in 
this way the contraction which the liquid may experience on 
entering is not considered, nor is the motion communicated to 
the surrounding water, which, up to a certain point, may give 
way before the surface A B, instead of crossing it. For these 
reasons it would be well in practice to admit a certain reduc- 
tion in the value of Q above given ; we could effect it, for ex- 
ample, by a co-efficient which we will value, at a rough esti- 
mate, at 0.80, for want of exact experiments on this subject. 

Here is an example for calculating Q, Let N = 12, ^ = 4, 
t' = 2^^.50, r'' = 3°^.0, H ^ 2^.0, S = 0^^-°^.17. We shall have 



^ = 0.8000, cos 
r 


-H.„.«.J, 


^ = 0.6667, cos 
r 


"';^ = »-=i'=^; 



/ COS -^5 = r'' cos ^~ = ^ (1.025 -|- 1.605) = 4.131 ; 

whence we deduce 

Q = 0^^.562, 



SPIRAL NOKIA. 125 

a number which would be reduced to O^^Ao about, by malti- 
plying by 0.80. 

As to the motive work to be expended in raising a certain 
weight P of water, it is composed ; 1st, of the work P H des- 
tined to overcome that of the weight ; 2d, of the work of fric- 
tion on the trunions and shoulders of the arbor O, which can 
be determined by means of known formulae ; 3d, of the work 
necessary to overcome the friction of the water against the 
solid walls with which it is in contact ; this work being very 
slight, if the involutes form tolerably large channels ; 4th, the 
work necessary to give to the water the absolute velocity with 
which it leaves the wheel. This last work will also be very 
slight, if we take care to make the wheel turn slowly ; for the 
lowest point of any involute whatever being always on the 
vertical through 1, we see that the water that has already en- 
tered the interior of the canal A B I G, and that which will 
still enter in the course of the same revolution, will only be 
completely emptied out after an entire revolution, reckoning 
from the position indicated by the figure. The water rises 
then with little absolute velocity into the machine, and conse- 
quently a small portion of the motive work is employed in 
giving to it an unproductive living force. But it must not be 
forgotten that this supposes slowness of revolution around the 
axis O. 

To sum up, we will calculate the first two portions of the 
motive work, which are the most important, and in order to 
account approximately for the other two, we will multiply the 
sum of the calculated portions by a co-efficient a little greater 
than unity. 

The first idea of the noria is very old, since Yitruvius speaks 
of a similar machine ; it was Lafaye who, in 1717, proposed 
giving it the form we have described above. This machine 



126 SPIRAL NOEIA. 

seems susceptible of a veij good delivery, and is well adapted 
to raising large volumes of water ; but the height to which the 
water is raised, always less than the radius of the wheel, is 
necessarily very limited ; besides, this wheel is heavy, and on 
this account hard to transport. 



CENTRIFUGAL PUMP. 



24. Lifting turbines ; centrifugal pump. — The greater part 
of the machines which are used to turn the motive power of a 
head of water to account, can, with a few changes, be converted 
into machines for raising water, and the reverse. Thus, for 
example, if a breast-wheel, set in a water-course, receives a 
motion about its horizontal axis, by the action of any motor, so 
that the floats may ascend the circular flume, these floats will 
carry up with them the water from the tail race and throw it 
into the head race : we would then obtain, in principle, the 
lifting wheel. In like manner, let us take one of Fom-neyron's 
turbines, and make the intervals between the directing parti- 
tions communicate directly with the tail race, and let the outer 
orifices of the turbine open into a compartment from which the 
ascent pipe leads ; when a motion of rotation is impressed on 
the wheel, the water contained in the floats will be urged to- 
ward the exterior by the centrifugal force, and will reach the 
enclosed compartment with an excess of pressure which will 
cause it to ascend the pipe to a certain height, the greater as 
the rotation becomes more rapid. If the pipe is not too high, 
a delivery of water will take place at its end ; and this, more- 
over, will be continuous, the water thrown out by the centrifu- 
gal force being incessantly replaced by that from the tail race, 
which tends to fill up the empty space between the partitions. 

The theory of such a turbine, which we might call a lifting 
turhine^ resembles very closely that of (No. 15). But as the 



128 



CENTEIFUGAL PTJMP. 



machine there discussed has not as yet been set up or experi- 
mented upon, it need no longer be dwelt upon. We will pass 
to the study of a pump called the centrifugal pump, which 
belongs to the same class of machines, but which bears, how- 
ever, a greater resemblance to reaction wheels. 

A wheel composed of a series of cylindrical floats, such as 
B C (Fig. 21), assembled between two annular plates, is caused 




Fig. 21. 



to turn around a horizontal axis projected at A. The water^ 
from the basin to be emptied, comes freely within the circle 
A B, which limits the floats on the inside, either because the 
centre A is a little below the level N N of this bay, or by 
means of suction pipes. The motion of rotation impressed on 
this wheel drives the water from the canals B C, B' C^, into the 
annular space D, where it acquires a pressure sufficient to drive it 
up the pipe E, the only means of escape open to it, and by which 
it reaches the upper basin. The angular velocity of the arbor 
A being known, as well as all the dimensions of the machine, 
and its position relatively to the basins of departure and arrival, 



CENTRIFUGAL PUMP. 129 

we can find the amount of water pumped up per second, the 
motive work that it requires, and its delivery. 

To show this, let us call 

H the difference of level between the two basins ; 

h the depth of the centre A below the level of the lower ; 

T the exterior radius A C of the wheel ; 

h the distance apart of the two annular plates, which confine 
the floats ; 

w the angular velocity of the arbor A ; 

^ the absolute velocity of the water when it leaves the floats ; 

u the velocity w t* at the outer circumference of the wheel ; 

w the relative velocity of the water at the same point ; 

y the acute angle formed by the velocities w and u — that is, 
the angle at which the floats cut the outer circumference ; 

J? the pressure of the water at its point of entrance into the 
interval between the floats ; 

jp' its pressure at the point of exit ; 

^a the atmospheric pressure ; 

n the weight of the cubic metre of water. 

We will begin by simplifying the question a little by means 
of a few hypotheses. First, we will neglect the absolute velo- 
city of the water in the ascent pipe and in the conduit which 
conveys it to the floats, which may be allowed if the cross sec- 
tions of these conduits are sufiiciently large relatively to the 
volume pumped out. The radius A B, however, should still 
be sufficiently small so that the velocity of rotation of the point 
B may be neglected ; in other words, we will consider the in- 
troduction of the water into the wheel as taking place along 
the axis, without any velocity occasioned by the motion, and 
consequently without any relative velocity. Secondly, we will 
conduct our argument as though the water were displaced hori- 
zontally in its passage across the wheel ; the height of this last 



130 CENTRIFUGAL PUMP. 

will be supposed slight relatively to H, so that it can be left 
out of account. Besides, nothing in practice would prevent 
our assuming the arbor A as vertical ; but this would be a 
matter of very little importance in the result. 

This granted, the pressure varying according to the hydro- 
static law from the lower basin to the point of entrance, and 
from the point of exit to the other basin, we will have 

whence, by subtraction, 

n 

Kow, if we apply Bernouilli's theorem to the relative motion 
of a molecule following the curve B C, the fictitious gain of 

head will be expressed by — — or — — , and we shall find 

w^ _ p —p' u^ 



2 p' n 2 ^' 

or else 

WJ'=: -2^H + t^^ ... (1) 

an equation giving w since u is known. This first result gives 
the means of calculating the amount Q pumped up in each 
second. In fact, the water leaving the floats cuts a cylindrical 
surface 2 -tt 5 r at an angle y and with the relative velocity w ; 
then the total orifice of exit, measured perpendicularly to ^^, is 
2 -jr 5 r sin 7, and consequently 

Q = 2 <»' J 7" ^(? sin 7. . . . (2) 
The motive work consumed per second in making the wheel 
turn includes first the work n Q H ; then the water reaching 
the annular space D with a velocity -y, this is lost in useless 

disturbance ; whence there results a molecular work n Q -r — . 

2 g 



CENTKIFUGAL PUMP. 131 

Thus then, throwing out of account the other frictions, the 

work expended per second will be n Q (H + ^ — ) ; and as the 

useful work is only n Q H, the effective delivery /a will have for 
its value 

. = ^-r=— V- • • • (^) 

There remains to determine v ; now v is the resultant of w and 
u^ hence we have 

v^ =1^ u^ -{- w^ -\- ^2, u w cos 7, 
or from eq. (1) 

v^ = - 2 ^ H + 2 '?^' - 2 ^^ cos 7 V — 2 g'R-\- u' (4) 

Equations (1), (2), (3), and (4) give the means of solving with- 
out difficulty the questions proposed. 

Let us again see by what means we could obtain the greatest 
possible result of the motive power. Expression (3) for the 
effective delivery becomes, substituting for v its value, and 

makmo^ — — — = x. 

1 



X — X cos 7 4/ /^2 _ 2 
we shall then have the maximum of f^, considered as a function 
of X, in seeking the minimum of the denominator, or, what 

amounts to the same, the minimum of -. We shall conduct 

this research as in (No. 21) ; we will write 
1 



a?'^ — - = a? cos 7 r cc^ — 2 



or, by making the radical disappear and transposing, 
X* sin'' y — 2 x^ ( cos' 7^ H — - = 0. 



9 



132 CENTRIFUGAL PTJMP. 

IS'ow M- can only receive values that, substituted in this biquad- 
ratic equation, will give ^ real and positive ; hence we have 

or successively 

1 2 

—: COS* V — _ cos* 7 4- cos* r > 0, 

l-? + cosV>0, 
(* ft 



(-:-■)■> 



sin 7. 



As sin 7 and - — 1 are positive quantities, we can extract the 
square root of both members and place 

— 1 > sin 7, or - > 1 + sin 7 ; 

the minimum of - has then for its value 1 + sin 7, and the 

limit of the effective delivery a, will be ; — . The corre- 

•^ ' 1 + sm 7 

spending value x^ of x is obtained from the above biquadratic 

equation, which gives 

1 

- — cos 7 ^ . - 

2 _|u,^ _ 1 -I- sm 7 — cos 7 _ 1 + sm 7 

' sin* 7 sin* 7 sin 7 

Thus the most favorable velocity u for the effective delivery is 

obtained from this equation 

2 -PP 1 + sin 7 

u = an. — i- : 

sm 7 ' 

/It 
whence we get the angular velocity w = -, and the number of 

revolutions per minute N = . The effective delivery being 



CENTRIFUGAL PUMP. 133 

then we may be tempted, in order to increase it, to 

1 -h sin 7 

make 7 very small ; but we see that the velocities u and w 

would become very great, and we should thus lose a great deal 

in the friction of the water against the floats. Besides, we 

have, from equations (1) and (2), 

Q zz: 2 -TT J /> sin 7 4/ 'w' — 2 ^ H = 2iflr V gH sin 7. 4/ a?' — 2] 

the amount Q', which corresponds to the maximum effective 

delivery, will then be 

Q' = 2 ^ 5 7' VJK. sin ^. W i-±-?i^— 2 

^ sm 7 



= 2 "n- J /• 1^ ^ H |/ sin 7 (1 — sin 7). 
This amount reduces to zero at the same time as 7 ; when 
7 alone varies, Q' becomes a maximum for sin 7 = 1 — sin 7, or 

sin 7 = - or 7 = 30° ; the effective delivery is then 

'^\ 

that is ^. The value 7 = <9 is consequently inadmissible as re- 

o 

ducing the amount to zero ; but from this point of view it is 
not well to go beyond y = 80\ On the other hand, this last 
value does not give a very high theoretical effective delivery ; 
perhaps the best thing to do in practice would be to take 7 be- 
tween 15 and 20 degrees. For 7 = 15°, for example, the effec- 
tive delivery increases to tt^^^w^ = 0.794, and the product 

1 -j" 0.2588 



V sin 7 (1 — sin 7) is decreased to 0.438, whereas it is 0.50 for 
y = 30° ; it is a diminution that could be compensated for by 
a slight increase of r or h. 

Instead of arranging the wheel as represented in Fig. 21, we 
might adopt two separate canals, as in Fig. 19. In this case 
the expression for the amount would change, and the angle 7 



134 CENTKIFUGAL PUMP. 

might become zero ; but, whereas u cannot increase to infinity, 

the value ; — would no lonarer be admissible for — =, and 

sih 7 g H 

we should have to depart more or less from the limit of the 

effective delivery. Furthermore, for an equal expenditure, we 

should probably lose more in friction. 



AUTHORITIES ON WATER WHEELS. 

Experiences sur les Roues Hydrauliques a aubes planeSy et sur les Roues 
Hydrauliques a augets, by Morin. 

Experiments made by the Committee of the FrankHn Institute on Water 
Wheels, in 1829-30. See Journal of the Franklin Institute, Sd Series, Vol. 
I., pp. 149, 154, &c., and Yol. II., p. 2. 

Experiments on Water Wheels, by Elwood Morris. See Jour. Frank. 
Ins., Sd Series, Vol. IV., p. 222. 

Memoir e sur les Roues Hydrauliques a aubes courhes, mues par dessous, by 
Poncelet. 

Experiences sur les Roues Hydrauliques a axe vertical, appelees Turbines, by 
Morin. 

Experiments on the Turbines of Fourneyron, by Elwood Morris. See 
Jour. Frank. Ins., Dec. 1843, and Jour. Frank. Ins., 3c? Series, Vol. IV., p. 
303. 

Lowell HydrauHc Experiments, 2d Ed., 1868, by Jas. B. Francis. 



APPENDIX 



Comparative Table of French and United States Measures, 

Pounds avoirdupois in a kilogramme 2.2 

Inch in a millimetre 0.039 

Inch in a centimetre 0.393 

Inch in a decimetre 3.937 

Feet in a metre 3.280 

Yard in a metre. . ^ 1.093 

Square feet in a square metre 10.7643 

Cubic inch in a cubic centimetre 0.061 

Cubic feet in a cubic metre 35.316 

Quart in a litre 1.0567 

IS'oTE, — A cubic metre of distilled water weighs one thousand 
kilogrammes. 

The litre contains one cubic decimetre of distilled water, and 
weighs one kilogramme. 

The horse-power of the French is 75 kilogrammetres, equiva- 
lent to 642|- foot-pounds per second ; the English horse-power 
being 550 foot-pounds per second. The value of g is 9.81 
metres; and that of the height of a column of water equivalent 
to the atmospheric pressure is taken at 10.33 metres. 



Note A. Art. 1. 

The formula given in this article is to be found, with the 
exception of variations in the notation, in all works of applied 



136 APPENDIX. 

mechanics in which the subject of the theory of machines is 
discussed (see, for example, Moseley's Engineering and Archi- 
tecture, Am. Ed., p. 146). The only term in it which is not 
generally found in other works is the one (H — H,,) 2 m ^, which 
expresses the work expended in overcoming the weight of any 
part of the machine, when its centre of gravity is raised from 
one level to another, represented by the vertical height (H — 
Hq), and the corresponding work by (H — H^) 2 m p' / as, for 
example, in the case of a wheel revolving on a horizontal axle, 
the axis of which does not coincide with its centre of gravity ; 
or in that of a revolving crank ; in both of which cases the 
work expended will be equal to the product of the weight 
1 mg raised, and the vertical height (H — H,,) passed over by 
its centre of gravity. But, in all like cases, as in the descent of 
the centre of gravity from its highest to its lowest position, the 
same amount of work will be restored by the action of gravity, 
the total work expended will be zero for each revolution, and 
the term (H — HJ ^mg will disappear from the formula. 



Note B. Art. 2. 

1 XJ' — U' 
The equation H C4+ t)) = ^ is the modified 

^ 2p 

form of what is known as Bernouilli's theorem as applied to 
the case treated of in Art. 2. 

This theorem, applied to the phenomena of the flow of a 
heavy homogeneous fluid, may be generally thus stated: The 
increase of height due to the velocity is equal to the difference 
hetween the effective head and the loss of head. 

In this case H is the efiective head : (4 4- t^ is the loss 

9 

of head from the dynamical effect ^ imparted to the wheel by 



APPENDIX. 137 

the action of the water, and the work tj due to the various sec- 

XJ'' — U' 
ondarj resistances ; and the term ^ the height due to 

the velocity. 

Multiplying each member of the above equation by g^ we 
obtain 

or, in other words, the modified expression of the general for- 
mula Note A, as applied to this case. 

See Bresse. Mecanique Ajpjpliquee. Vol. 2, JVos. 12 and 



Note C. Art. 4. 
In the equation 

9 2 

which expresses the force applied horizontally at the centre of 
the submerged portion of the bucket, the second term of the 

second member 77 n 5 (h''^ — A'*), represents the diminution of 

2 

the force imparted to the wheel by the current, arising from 
the increase of depth of the water as it leaves the wheel, or by 
the back water ; or, in other words, the difference of level be- 
tween the point C, before the depth of the current is affected 
by the action of the wheel, and the point E, where the deptli 
of the current has increased from the back water. This differ- 
ence of level receives the name of a surface fall. 

The relations existing between the two terms of the second 
member of the equation, leaving out of consideration the action 
of the wheel, may be established in the following manner. 

Considering the portion of the current comprised between the 



138 



APPENDIX. 



two sections C B, E F (Fig. 3), at a short distance apart, be- 
tween which the surface fall takes place, we can apply to the 
liquid system C B E F, comprised between these sections, the 
theorem of the quantities of motion projected on the axis of 
the current, which, in the present case, may be regarded as 
horizontal. IN^ow, during a very short interval of time ^, the 
system C B E F will have changed its position to C B' E' F', 
and, in virtue of the supposed permanency of the motion, each 




Fig. 8. 



point of the intermediate portion C B' E F will have equal 
masses moving with the same velocity at the beginning and 
ending of the time ^ ; the variation in the projected quantity of 
motion of the system C B E F, during the time ^, will therefore 
be equal to the quantity of motion of the portion included be- 
tween the final sections E F, E' F', and that comprised between 
the initial sections C B, C B'. 

To find these quantities of motion. Represent by w' a supei-- 
ficial element of the section E F, and by v' the velocity of the 
fluid thread which flows through it ; v' 6 will then be the length 
of this thread for the time ^, between the sections E F and 
E' F' ; and w' v' & will be the volume of the thread which has 
u for its base and v' 6 for its length. Representing by n the 

weight of a cubic metre of the liquid, — (*i' v' 6 will be the cor- 
responding mass of this volume, and — w' v^^ 6 its quantity of 
motion ; and, designating by s the sum of all the elements w', 



APPENDIX. 139 

— s w' v'^ 6 will be the quantity of motion of the portion 
9 

E F ET^ of the liquid comprised between the two final sec- 
tions. In like manner, v being the velocity with which each 
thread flows through an element w of the section B C, the quan- 
tity of motion of the portion of liquid between the sections B C 

and B' C^ will be expressed by — s w -y^^. The increase, there- 

fore, in the quantity of motion during the time 6 will be ex- 
pressed by 

9 

But as V and v' may be assumed as sensibly equal to the mean 

velocities of the current in sections E F, C B, then s ^' v' and 
s w 'y will be the volumes corresponding to these velocities; and 

— ^ (ji' v' and — 2 w 'y the corresponding: masses. But since. 

9 g 

from the permanency of the motion w^ ^y^ = w v^ the expression 
for the increase of the quantity of motion, for the time & will 
therefore take the form 

in which P represents the weight of the water expended in each 

p 

second, and — its corresponding mass. 
tf 
The expression here found is equal to the sum of the impul- 
sions of the forces exterior to the liquid system considered 
during the time ^, also projected on the horizontal axis of the 
current. From the form given to the section of the race, which 
is rectangular, the direction of the axis of the current, which is 
assumed as horizontal, between the extreme sections, and the 
short distance between these sections, the only impulsions of the 
pressures upon the liquid system are those on the sections E F 



140 APPENDIX. 

and C B. Eepresenting, then, bj h the breadth of the sections, 
bj h' and h their respective depths, their respective areas will 
be expressed by h h' and hh ; the pressures on these areas will 

be n 5 A' X - A^ and nhh x -h ; and for the respective pro- 
jected sums of the impulsions of these pressures, during the 

time ^, we shall have -Hh ^ h'"^ and - lih & h^^ since from the 

circumstances of the motion the pressures follow the hydro- 
static law. The impulsion in the direction of the motion will 

therefore be expressed by - n 5 ^ (A^ — h'"")^ from which we ob- 
tain 

to express the relation in question. 

See Bresse. Meccmique Ajpjpliquee. Vol. 2, No. 83^^. 245. 



Note D. AH. 9. 

The term Cfl — ^) — c, which expresses the arc inter- 

cepted between two buckets, taken at the middle point of their 
depth, is obtained as follows : 

R being the exterior radius of the wheel, corresponding to 
the arc C, the radius of the arc at the middle point of the 

bucket will be R — - _p ; calling x the arc corresponding to 

this radius, we have 

E:E-1^::0:.. .-..= 0(1-^^); 

and for the arc intercepted between the two buckets at their 
middle point the expression above. 



APPENDIX. 141 



mte'S.. 



By the courteous permission of James B. Francis, Esq., granted through 
Gen. John C. Palfrey, the following extracts were taken from the val- 
uable work of Mr. Francis, under the title of " Lowell Hydraulic Experi- 



A YAST amount of ingenuity has been expended by intel- 
ligent millwrights on turbines ; and it was said, several years 
since, that not less than three hundred patents relating to them 
had been granted by the United States Government. They 
continue, perhaps, as much as ever to be the subject of almost 
innumerable modifications. Within a few years there has been 
a manifest improvement in them, and there are now several 
varieties in use, in which the wheels themselves are of simple 
forms, and of single pieces of cast iron, giving a useful effect 
approaching sixty per cent, of the power expended. 

In the journal of the Frauklin Institute, Mr. Morris also 
published an account of a series of experiments, by himself, on 
two turbines constructed from his own designs, and then ope- 
rating in the neighborhood of Philadelphia. 

The experiments on one of these wheels indicate a useful 
effect of seventy-five per cent, of the power expended, a result 
as good as that claimed for the practical effect of the best over- 
shot wheels, which had heretofore in this country been con- 
sidered unapproachable in their economical use of water. 

In the year 1844, Uriah A. Boyden, Esq., an eminent 
hydraulic engineer of Massachusetts, designed a turbine of 
about seventy-five horse power, for the picking-house of the 
Appleton Company's cotton-mills, at Lowell, in Massachusetts, 
in which wheel Mr. Boyden introduced several improvements 
of great value. 

The performance of the Appleton Company's turbine was 



142 APPENDIX. 

carefully ascertained bj Mr. Bojden, and its effective power, 
exclusive of that required to carry the wheel itself, a pair of 
bevel gears, and the horizontal shaft carrying the friction-pulley 
of a Prony dynamometer, was found to be seventy-eight per 
cent, of the power expended. 

In the year 1846, Mr. Boyden superintended the construc- 
tion of three turbines, of about one hundred and ninety horse- 
power each, for the same company. By the terms of the con- 
tract, Mr. Boyden 's compensation depended on the performance 
of the turbines ; and it was stipulated that two of them should 
be tested. In accordance with the contract, two of the turbines 
were tested, a very perfect apparatus being designed by Mr. 
Boyden for the purpose, consisting essentially of a Prony dyna- 
mometer to measure the useful effects, and a weir to gauge the 
quantity of water expended. 

The observations were put into the hands of the author for 
computation, who found that the mean maximum effective 
power for the two turbines tested was eighty-eight per cent, of 
the power of the water expended. 

According to the terms of the contract, this made the com- 
pensation for engineering services, and patent rights for these 
three wheels, amount to fifty-two hundred dollars, which sum 
was paid by the Appleton Company without objection. 

These turbines have now been in operation about eight 
years, and their performance has been, in every respect, entirely 
satisfactory. The iron work for these wheels was constructed 
by Messrs. Gay & Silver, at their machine-shop at J^orth 
Chelmsford, near Lowell ; the workmanship was of the finest 
description, and of a delicacy and accuracy altogether unpre- 
cedented in constructions of this class. 

These wheels, of course, contained Mr. Boyden 's latest im- 
provements, and it was evidently for his pecuniary interest that 



APPENDIX. 143 

the wheels slionld be as perfect as possible, without much regard 
to cost. The principal points in which one of them differs 
from the constructions of Fournejron are as follows : — 

The wooden flume conducting the water immediately to the 
turbine is in the form of an inverted truncated cone^ the water 
heing introduced into the upper part of the cone^ on one side of the 
axis of the cone {which coincides with the axis of the turbine)^ in 
such a manner that the water ^ as it descends in the cone^ has a 
gradually increasing velocity and a spiral motion / the horizontal 
component of the spiral motion being in the direction of the 
motion of the wheel. This horizontal motion is derived from 
the necessary velocity with which the water enters the trun- 
cated cone; and the arrangement is such that, if perfectly pro- 
portioned, there would be no loss of power between the nearly 
still water in the principal penstock and the guides or leading 
curves near the wheel, except from the friction of the water 
against the walls of the passages. It is not to be supposed that 
the construction is so perfect as to avoid all loss, except from 
friction ; but there is, without doubt, a distinct advantage in 
this arrangement over that which had been usually adopted, 
and where no attempt had been made to avoid sudden changes 
of direction and velocity. 

The guides^ or leading cicrves (Figs. A, B), are not perpen- 
dicular^ but a little inclined bachvjards from the 'motion of the 
wheel, so that the water, descending with a spiral motion, meets 
only the edges of the guides. This leaning of the guides has 
also another valuable effect : when the regulating gate is raised 
only a small part of the height of the wheel, the guides do not 
completely fulfil their office of directing the water, the water 
entering the wheel more nearly in the direction of the radius 
than when the gate is fully raised ; by leaning the guides it will 
be seen the ends of the guides near the wheel are inclined, the 



14:4: APPENDIX. 

bottom part standing farther forward, and operating mure 
efficiently in directing the water when the gate is partially 
raised, than if the guides were perpendicular. 

In Fourneyron's constructions a garniture is attached to 
the regulating gate, and moves with it, for the purpose of di- 
minishing the contraction. This, considered apart from the 
mechanical difficulties, is probably the best arrangement ; to 
be perfect, however, theoretically, this garniture should be of 
different forms for different heights of gate ; but this is evi- 
dently impracticable. 

In the Apjpleton Uirhine the garniture is attached to the 
guides^ the gate {at least the lower part of it) being a simple thin 
cylinder. By this arrangement the gate meets with much 
less obstruction to its motion than in the old arrangement, un- 
less the parts are so loosely fitted as to be objectionable ; and it 
is believed that the coefficient of effect, for a partial gate, is 
proportionally as good as under the old arrangement. 

On the outside of the wheel is fitted an apparatus, named hy 
Mr, Boyden the Diffuser, The object of this extremely inter- 
esting invention is to render useful a part of the power other- 
wise en,tirely lost, in consequence of the water leaving the wheel 
with a considerable velocity. It consists, essentially, of two 
stationary rings or discs, placed concentrically with the wheel, 
having an interior diameter a very little larger than the exte- 
rior diameter of the wheel ; and an exterior diameter equal to 
about twice that of the wheel ; the height between the discs at 
their interior circumference is a very little greater than that of 
the orifices in the exterior circumference of the wheel, and at 
the exterior circumference of the discs the height between them 
is about twice as great as at the interior circumference ; the form 
of the surfaces connecting the interior and exterior circumfer- 
ences of the discs is gently rounded, the first elements of the 



APPENDIX. 145 

curves near the interior circumferences being nearly horizon- 
tal. Tliere is consequently included between the two surfaces 
an aperture gradually enlarging from the exterior circumference 
of the wheel to the exterior circumference of the diffuser. When 
the regulating gate is raised to its full height, the section through 
which the water passes will be increased, by insensible degrees, 
in the proportion of one to four, and if the velocity is uniform 
in all parts of the diffuser at the same distance from the wheel, 
the velocity of the water will be diminished in the same pro- 
portion ; or its velocity on leaving the diffuser will be one-fourth 
of that at its entrance. By the doctrine of living forces, the 
power of the water in passing through the diffuser must, there- 
fore, be diminished to one-sixteenth of the power at its entrance. 
It is essential to the proper action of the diffuser that it should 
be entirely under water, and the power rendered useful by it 
is expended in diminishing the pressure against the water issu- 
ing from the exterior orifices of the wheel ; and the effect pro- 
duced is the same as if the available form under which the 
turbine is acting is increased a certain amount. It appeai-s 
probable that a diffuser of different proportions from those above 
indicated would operate with some advantage without being 
submerged. It is nearly always inconvenient to place the 
wheel entirely below low- water mark ; up to this time, however, 
all that have been fitted up with a diffuser have been so placed ; 
and indeed, to obtain the full effect of a fall of water, it appears 
essential, even when a diffuser is not used, that the wheel should 
be placed below the lowest level to which the water falls in the 
wheel-pit, when the wheel is in operation. 

The action of the diffuser depends upon similar principles 
to that of diverging conical tubes, which, when of certain pro- 
portions, it is well known, increase the discharge ; the author 
has not met with any experiments on tubes of this form dis- 



146 APPENDIX. 

charging under water although there is good reason to believe 
that tubes of greater length and divergency would operate more 
effectively under water than when discharging freely in the 
air, and that results might be obtained that are now deemed 
impossible by most engineers. 

Experiments on the same turbine, with and without a dif- 
fuser, show a gain in the coefficient of effect, due to the latter, 
of about three per cent. By the principles of living forces, and 
assuming that the motion of the water is free from irregularity, 
the gain should be about five per cent. The difference is due, 
in part at least, to the unstable equilibrium of water flowing 
through expanding apertures ; this must interfere with the uni- 
formity of the velocities of the fluid streams, at equal distances 
from the wheel. 

Suspending the wheel on the top of the vertical shaft (Fig. 
A), instead of running it on a step at the bottom. This had been 
previously attempted, but not with such success as to warrant 
its general adoption. It has been accomplished with complete 
success by Mr. Boyden, whose mode is to cut the upper part of 
the shaft into a series of necks, and to rest the projecting parts 
upon corresponding parts of a box. A proper fit is secured by 
lining the box, which is of cast-iron, with Babbitt metal — a soft 
metallic composition consisting, principally, of tin ; the cast- 
iron box is made with suitable projections and recesses, to sup- 
port and retain the soft metal, which is melted and poured into 
it, the shaft being at the same time in its proper position in the 
box. It will readily be seen that a great amount of bearing- 
surface can be easily obtained by this mode, and also, what is 
of equal importance, it may be near the axis ; the lining metal, 
being soft, yields a little if any part of the bearing should re- 
ceive a great excess of weight. The cast-iron box is suspended 
on gimbals, similar to those usually adopted for mariners' com- 



APPENDIX. 147 

passes and clironoineters, which arrangement permits the box 
to oscillate freelj in all directions, horizontally, and prevents, 
in a great measure, all danger of breaking the shaft at the 
necks, in consequence of imperfections in the workmanship or 
in the adjustments. Several years' experience has shown that 
this arrangement, carefully constructed, is all that can be de- 
sired ; and that a bearing thus constructed is as durable, and 
can be as readily oiled and taken care of, as any of the ordinary 
bearings in a manufactory. 

The buckets are secured to the crowns of the wheel in a 
novel and much more perfect manner than had been previously 
used ; the crowns are first turned to the required form, and 
made smooth ; by ingenious machinery designed for the pur- 
pose, grooves are cut with great accuracy in the crowns, of the 
exact curvature of the buckets ; mortices are cut through the 
crowns in several places in each groove ; the buckets, or floats, 
are made with corresponding tenons, which project through the 
crowns, and are riveted on the bottom of the lower crown, and 
on the top of the upper crown ; this construction gives the re- 
quisite strength and firmness, with buckets of much thinner 
iron than was necessary under any of the old arrangements ; it 
also leaves the passages through the wheel entirely free from 
injurious obstructions. 

In the year 1849, the manufacturing companies at Lowell 

purchased of Mr. Boyden the right to use all his improvements 

relating to turbines and other hydraulic motors. Since that 

time it has devolved upon the author, as the chief engineer of 

these companies, to design and superintend the construction of 

such turbines as might be wanted for their manufactories, and 

to aid him in this important undertaking, Mr. Boyden has 

communicated to him copies of many of his designs for 

turbines, together with the results of experiments upon a por- 
10 



148 APPENDIX. 

tion of them ; he has communicated, however, but little theo- 
retical information, and the author has been guided principally 
by a comparison of the most successful designs, and such light 
as he could obtain from writers on this most intricate subject. 

Summary description of one of the turbines at the Tremont 
Jfills, Lowell. Figs. A, B, C. 

Fig. A is a vertical section of the turbine through the axis of the wheel 
shaft ; Fig. B is a portion of the plan, on an enlarged scale, showing the 
disposition of the leading curves and buckets and diffuser ; Fig. C is a cross 
section of the wheel and diffuser on an enlarged scale, and the more adja- 
cent parts. The letters on the corresponding parts of the figures are the 
same. 

The water is conveyed to the wheel of the turbine, from the 
forebay by a supply pipe, the greater portion of which, from 
the forebay downwards, is of wrought iron, and of gradually 
diminishing diameter towards the lower portion I, termed the 
curbs ^ which is of cast iron. The curbs are supported on col- 
umns, which rest on cast-iron supports firmly imbedded in the 
wheel-pit. 

The Disc K, K', K'^, to which the guides for the water, or 
the leading curves, thirty -three in number, are attached, is sus- 
pended from the upper end of the cast-iron curb, by means of 
the disc-pipes M M. 

The leading curves are of Russian iron, one-tenth of an inch 
in thickness. The upper corners of these, near the wheel, are 
connected by what is termed the garniture L, L', L'^, intended 
to diminish the contraction of the fluid vein when the regulat- 
ing gate is fully raised. 

The disc-pipe is very securely fastened, to sustain the pressure 
of the water on the disc. The escape of water, between the 
upper curb and the upper flange of the disc-pipe, is prevented 
by a band of leather on the outside, enclosed w^ithin an iron 



APPENDIX. 149 

ring. This pipe is so fastened as to prevent its rotating in a 
direction opposite to that in which the water flows out. 

The regulating gate is a cast-iron cylinder, R, enclosing the 
disc and curves, and which, raised or lowered by suitable 
machinery, regulates the amount of water let on the wheel B 
B' B'', exterior to it. 

The wheel consists of a central plate of cast-iron and two 
crowns, C,C, C, C^^, of the same material to which the buckets 
are attached. These pieces are all accurately turned, and pol- 
ished, to offer the least obstruction in revolving rapidly in the 
water. 

The huckets^ made of Russian iron, are forty-four in number, 
and each -^ of an inch thick. They are firmly fastened to the 
crowns. 

The vertical shaft D, from which motion is communicated 
to the machinery by suitable gearing, is of wrought-iron. In- 
stead of resting on a gudgeon, or step at bottom, it is suspended 
from a suspension box, E', by which the collars at the top are en- 
closed. These collars are of steel, and are fastened to the upper 
portion of the shaft, which last can be detached from the lower 
portion. 

The suspension hox is lined with Babbitt metal, a soft compo- 
sition consisting mostly of tin, and capable of sustaining a 
pressure of from 50 lbs. to 100 lbs. per square inch, without 
sensible diminution of durability. The box consists of two 
parts, for the convenience of fastening it on, or the reverse. 
The box rests upon the gimbal G, which is so arranged that the 
suspension box, the shaft, and the wheel can be lowered or 
raised, and the suspension box be allowed to oscillate laterally, 
so as to avoid subjecting it to any lateral strain. 

The lower end of the shaft has a cast-steel pin, O, fixed to it. 
This is retained in its place by the step, which is made of three 



150 APPENDIX. 

parts, and lined with case-hardened iron. The step can be ad- 
justed by horizontal screws, by a small lateral motion given by 
them to it. 

Rules for jprojportioning turbines. In making the designs 
for the Tremont and other turbines, the author has been guided 
by the following rules, which he has been led to, by a compari- 
son of several turbines designed by Mr. Boyden, which have 
been carefully tested, and found to operate well. 

Kule 1st. The sum of the shortest distances between the 
buckets should be equal to the diameter of the wheel. 

Rule 2d. The height of the orifices of the circumference of 
the wheel should be equal to one-tenth of the diameter of the 
wheel. 

Rule 3d. The width of the crowns should be four times the 
shortest distance between the buckets. 

Rule 4th. The sum of the shortest distances between the 
curved guides, taken near the wheel, should be equal to the in- 
terior diameter of the wheel. 

The turbines, from a comparison of which the above rules 
were derived, varied in diameter from twenty-eight inches to 
nearly one hundred inches, and operated on falls from thirty 
feet to thirteen feet. The author believes that they may be 
safely followed for all falls between five feet and forty feet, and 
for all diameters not less than two feet ; and, with judicious 
arrangements in other respects, and careful workmanship, a 
useful effect of seventy-five per cent, of the power expended 
may be relied upon. For falls greater than forty feet, the sec- 
ond rule should be modified, by making the height of the 
orifices smaller in proportion to the diameter of the wheel. 

Taking the foregoing rules as a basis, we may, by aid of the 
experiments on the Tremont turbine, establish the following 
formulas. Let 



APPENDIX. 161 

D = the diameter of the wheel at the outer extremities of 
the buckets. 

d = the diameter at their inner extremities. 

H = the height of the orifices of discharge, at the outer ex- 
tremities of the buckets. 

W = width of crowns of the buckets. 

iT = the number of buckets. 

71 = the number of guides. 

P =z the horse-power of the turbine, of 550 ^^^ ^ 

A = the fall acting on the wheel. 

Q r= the quantity of water expended by the turbine, in cubic 
feet per second. 

Y = the velocity due the fall acting on the wheel. 

y = the velocity of the water passing the narrowest sections 
of the wheel. 

V = the velocity of the interior circumference of the wheel ; 
all velocities being in feet per second. 

C = the coefficient of Y\ or the ratio of the real velocity 
of the water passing the narrowest sections of the wheel, to the 
theoretical velocity due the fall acting on the wheel. 

The unit of length is the English foot. 

It is assumed that the useful eifect is seventy-five per cent. 
of the total power of the water expended. 

According to Rule 1st, we have the sum of the widths of the 
orifices of discharge, equal to D. Then the sum of the areas 
of all the orifices of discharge is equal to D H, 

By the fundamental law of hydraulics, we have 

Therefore Y' = C V~^H. 

For the quantity of water expended we have. 
Q = HD Y' = HD C VY^, 
From the extremely interesting and accurate experiments of 



152 APPENDIX. 

Mr. Francis on the expenditure of water by one of the Tremont 
wheels, recorded in his work, the following data are obtained 
from it : — 

For the sum of the widths of the orifices of discharge, 

U X 0.18757 = 8.25308 feet. 
Q = 138.1892 cubic feet per second; 
A = 12.903 feet ; 
V~2g = 8.0202 feet. 

Substituting these numerical results in the preceding value 
of Q, there obtains 

138.1892 = 7.68692 x 8.0202 Vl2:903 O, 
hence 

C = 0.624. 
By Rule 2d we have 

B:= 0.10 i>, hence BD = 0.10 D% 
hence Q= HDV = 0.10 D' C VTg'h. 

Calling the weight of a cubic foot of water 62.33 lbs., we 
have 

550 
or, substituting for Q the value just found, 
F = 0.0425 D' h VT, 
hence 

i> = 4.85 ./Z. 

The number of buckets is to a certain extent arbitrary, and 
would usually be determined by practical considerations. Some 
of the ideas to be kept in mind are the following : 

The pressure on each bucket is less, as the number is greater ; 
the greater number will therefore permit of the use of thinner 
iron, which is important in order to obtain the best results. 
The width of the crowns will be less for a greater number of 



APPENDIX. 15-3 

buckets. A narrow crown appears to be favorable to the useful 
effect, when the gate is only partially raised. As the spaces 
between the buckets must be proportionally narrower for a 
larger number of buckets, the liability to become choked up, 
either with anchor ice or other substances, is increased. The 
amount of power lost by the friction of the water against the 
surfaces of the buckets will not be materially changed, as the 
total amount of rubbing surface on the buckets will be nearly 
constant for the same diameter ; there will be a little less on 
the crown, for the larger number. The cost of the wheel will 
probably increase with the number of buckets. The thickness 
and quality of the iron, or other metal intended to be used for 
the buckets, will sometimes be an element. In some water 
wrought iron is rapidly corroded. 

The author is of opinion that a general rule cannot be given 
for the number of buckets ; among the numerous turbines work- 
ing rapidly in Lowell, there are examples in which the shortest 
distance between the buckets is as small as 0.75 of an inch, and 
in others as large as 2.75 inches. 

As a guide in practice, to be controlled by particular circum- 
stances, the following is proposed, to be limited to diameters 
of not less than two feet : — 

]^= 3(i> + 10). 
Taking the nearest whole nuraber for the value of N. 

The Tremont turbine is 8^ in diameter, and, according to the 
proposed rule, should have tifty-five buckets instead of furty- 
four. With fifty-five buckets, the crowns should ha\e a \\idtli 
of 7.2 inches instead of 9 inches. With the narrower width, it 
is probable that the useful effect, in proportion to the power 
expended, would have been a little greater when the gate was 
partially raised. 

By the 3d rule, we have for the width of the crowns, 



154: , APPENDIX. 



and for the interior diameter of the wheel, 

8i> 



d = D 



]sr 



Bv the 4th rule, d is also equal to the sum of the shortest 
distances between the guides, where the water leaves them. 

The number n of the guides is, to a certain extent, arbitrary. 
The practice at Lowell has been, usually, to have from a half to 
three-fourths of the number of the buckets ; exactly half would 
probably be objectionable, as it would tend to produce pulsa- 
tions or vibrations. 

The proper velocity to be given to the wheel is an impor- 
tant consideration. Experiment 80 (the one above used for 
data) on the Tremont turbine gives the maximum coefficient 
of effect of that wheel ; in that experiment, the velocity of the 
interior circumference of the wheel is 0.62645 of the velocity 
due to the fall acting on the wheel. By reference to other ex- 
periments, with the gate fully raised, it will be seen, however, 
that the coefficient of effect varies only about two per cent, 
from the maximum, for any velocity of the interior circumfer- 
ence, between fifty per cent, and seventy per cent, of that due 
to the fall acting upon the wheel. By reference to the experi- 
ments in which the gate is only partially raised, it will be seen 
that the maximum corresponds to slower velocities ; and as tur- 
bines, to admit being regulated in velocity for variable work, 
must, almost necessarily, be used with a gate not fully raised, 
it would appear proper to give them a velocity such that they 
will give a good effect under these circumstances. 

With this view, the following is extracted from the experi- 
ments in Table II. ; — 



APPENDIX. 



155 



Number of the ex- 
periment. 


Height of the regulat- 
ing gate in inches. 


Ratio of the velocity of the in- 
terior circumference of the 
Trheel, to the velocity due the 
fall acting upon the wheel cor- 
responding to the maximum 
coefl&cient of effect. 


30 
62 
73 

84 


11.49 

8.55 
5.65 
2.875 


0.62645 
0.56541 
0.56205 
0.48390 



Bj this table it would appear that, as turbines are generally 
used, a velocity of the interior circumference of the wheel, of 
about fifty- six per cent, of that due to the fall acting upon the 
wheel, would be most suitable. By reference to the diagram 
at Plate YI,"^ it will be seen that at this velocity, when the 
gate is fully raised, the coefficient of effect will be within less 
than one per cent, of the maximum. 

Other considerations, however, must usually be taken into ac- 
count in determining the velocity ; the most frequent is the vari- 
ation of the fall under which the wheel is intended to operate. 
If, for instance, it were required to establish a turbine of a given 
power on a fall liable to be diminished to one-half by back- 
water, and that the turbine should be of a capacity to give the 
requisite power at all times, in this case the dimensions of the 
turbine must be determined for the smallest fall ; but if it has 
assigned to it a velocity, to give the maximum effect at the 
smallest fall, it will evidently move too slow for the greatest 
fall, and this is the more objectionable, as, usually, Avhen the 
fall is greatest the quantity of water is the least, and it is of 
the most importance to obtain a good effect. It would then be 

* "Lowell Hydraulic Experiments." 



156 APPENDIX. 

usually the best arrangement to give the wheel a velocity cor- 
responding to the maximum coefficient of effect, when the fall 
is greatest. To assign this velocity, we must find the propor- 
tional height of the gate when the fall is greatest ; this may be 
determined approximately by aid of the experiments on the 
Tremont turbine. 

We have seen that P= 0.085 QTi. 

Now, if h is increased to 2 A, the velocity, and consequently 
the quantity, of water discharged will be increased in the pro- 
portion of VA to '/2A ; that is to say, the quantity for the fall- 
2A will be |/2 Q. 

Calling P' the total power of the turbine on the double fall, 
we have 

P'=0.085 4/2^2A, 
or, 

P'rr:0.085x 2.8284 § A. 

Thus, the total power of the turbine is increased 2.8284 times, 
by doubling the fall ; on the double fall, therefore, in order to 
preserve the effective power uniform, the regulating gate must 
be shut down to a point that will give only ^.-g-^-g-j part of the 
total power of the turbine. 

In Experiment 15, the fall acting upon the wheel was 12.888 
feet, and the total useful effect of the turbine was 85625.3 
lbs. raised one foot per second ; ^.-gVsr P^^^ ^^ ^^^ ^^ 30273.4 
lbs. ; consequently the same opening of gate that would give 
this last power on a fall of 12.888 feet, would give a power of 
85625.3 lbs. raised one foot per second, on a fall of 2x12.888 
feet= 25.776 feet. To find this opening of gate, we must have 
recourse to some of the other experiments. 

In Experiment 73, the fall was 13.310 feet, the height of the 
gate 5.65 inches, and the useful effect 58830.1 lbs. In Ex- 
periment 83 the fall was 13.435 feet, the height of the gate 



APPENDIX. 157 

2.875 inches, and the useful effect 27310.9 lbs. Heducing 
both these useful effects to what they would have been if the 
fall was 12.888 feet, 

the useful effect in experiment 73, 58830.l(3:|:|?^)*= 560545; 



83, 27310.9(^^4^")' = 25660.1. 
' \13.435/ 



By a comparison of the useful effects with the corresponding 
heights of gate, we find, by simple proportion of the differences, 
that a useful effect of 30273.4 lbs. raised one foot high per 
second, would be given when the height of the regulating gate 
was 3.296 inches. 

By another mode ; — 
As 25660.1 : 2.875 :: 30273.4 : 2.875 xf^f:f =3.392 in., a little 
consideration will show that the first mode must give too little, 
and the second too much ; taking a mean of the two results, we 
have for the height of the gate, giving ^.-gVsr ^^ ^^^^ total power 
of the turbine, 3.344 inches. Referring to Table II., we see 
that, with this height of gate, in order to obtain the best coeffi- 
cient of useful effect, the velocity of the interior circumference 
of the wheel should be about one-half of that due to the fall 
acting upon the wheel ; and by comparison of Experiments 74 
and 84, it will be seen that, with this height of gate and with 
this velocity, the coefficient of useful effect must be near 0.50. 

This example shows, in a strong light, the well-known defect 
of the turbine, viz., giving a diminished coefficient of useful 
effect at times when it is important to obtain the best results. 
One remedy for this defect would be, to have a spare turbine, 
to be used when the fall is greatly diminished ; this arrangement 
would permit the principal turbine to be made nearly of the di- 
mensions required for the greatest fall. As at other heights of 
the water economy of water is usually of less importance. 



158 



APPENDIX. 



the spare turbine might generally be of a cheaper construc- 
tion. 

To lay out the curve of the huckets^ the author makes use of 
the following method : — 

Keferring to Fig. D, the number of buckets, N, having been 

determined by the preceding rules, set off the arc GI —"^ — 

Let w = GH = I P', the shortest distance between the buckets ; 

t — the thickness of the metal forming the buckets. 

Make the arc G K = 5 w. Draw the radius O K, intersecting 
the interior circumference of the wheel at L ; the point L will 
be the inner extremity of the bucket. Draw the directrix L M 
tangent to the inner circumference of the wheel. Draw the 
arc O N, with the radius w + ^, from I as a centre ; the other 
directrix, G P, must be found by trial, the required conditions 
being, that, when the line M L is revolved round to the position 
G T, the point M being constantly on the directrix G P, and 
another point at the distance M G = P S, from the extremity 
of the line describing the bucket, being constantly on the di- 
rectrix M L, the curve described shall just touch the arc N O. 
A convenient line for a first approximation may be drawn by 
making the angle O G P = 11°. After determining the direc- 
trix according to the preceding method, if the angle O G P 
should be greater than 12°, or less than 10°, the length of the 
arc G K should be changed to bring the angle within these 
limits. 

The curve G S S' S'^ L, described as above, is nearly the quar- 
ter of an ellipse, and would be precisely so if the angle G M L 
was a right angle ; the curve may be readily described, me- 
chanically, with an apparatus similar to the elliptic trammel; 
there is, however, no difficulty in drawing it by a series of 
points, as is sufficiently obvious. 



APPENDIX. 159 

The trace adopted by the author for the corresponding guides 
is as follows : — 

The number n having been determined, divide the circle in 
which the extremities of the guides are found into n equal 
parts y W, W X, etc. 

Put w' for the width between two adjoining guides, 

and t' for the thickness of the metal forming the guides. 

d 
We have by Eule 4, w' = — 

With W as a centre, and the radius w' + t\ draw the arc 
Y Z ; and with X as a centre, and the radius 2(w' + f), draw 
the arc A^ B'. Through Y draw the portion of a circle, Y C, 
touching the arcs Y Z and A' B^ ; this will be the curve for the 
essential portion of the guide. The remainder of the guide, 
C D', should be drawn tangent to the curve C^ Y; a convenient 
radius is one that would cause the curve C D\ if continued to 
pass through the centre O. This part of the guide might be 
dispensed with, except that it affords great support to the part 
C Y, and thus permits the use of much thinner iron than would 
be necessary if the guide terminated at C^, or near it. 

Collecting together the foregoing formulas for proportioning 
turbines, which, it is understood, are to be limited to falls not 
exceeding forty feet, and to diameters not less than two feet, 
we have for the horse power, 

F = 0M25I)'h Vh ; 

for the diameter. 



D 



= ^-^^ \/S 



for the quantity of water discharged per second, 
Q = 0.5 D' y/h; 



160 APPENDIX. 

for the velocity of the interior circumference of the wheel, when 
the fall is not very variable, ^ 

V = 0.56 /2gli, 
or, i; = 4.491 |/A/ 

for the height of the orifices of discharge, 

^=0.10i>; 
for the number of buckets, 

ir=3 (i> + 10); 
for the shortest distance between two adjacent buckets, 

D 

for the width of the crown occupied by the buckets, 
for the interior diameter of the wheel, 

for the number of guides, 

71 = 0.50 iT to 0.75 iV^; 
for the shortest distance between two adjacent guides, 

u' = — • 

n 
Table has been computed by these formulas. 
For falls greater than forty feet, the height of the orifices in 
the circumference of the wheel should be diminished. The 
foregoing formulas may, however, still be made use of. Thus : 
supposing, for a high fall, it is determined to make the orifices 
three-fourths of that given by the formula; divide the given 
power, or quantity of water to be used, by 0.75, and use the 
quotient in place of the true power or quantity, in determining 
the dimensions of the turbine. !N"o modifications of the dimen- 
sions will be necessary, except that -^ of the diameter of the 



APPENDIX. 161 

turbine should be diminished to ^ of the diameter, to give the 
height of the orifices in the circumference. 

It is plain, from the method by which the preceding formulas 
have been obtained, that they cannot be considered as estab- 
lished, but should only be taken as guides in practical applica- 
tions, until some more satisfactory are proposed, or the intrica- 
cies of the turbine have been more fully unravelled. The 
turbine has been an object of deep interest to many learned 
mathematicians, but up to this time the results of their investi- 
gations, so far as they have been published, have afforded but 
little aid to hydraulic engineers. 

Diffiiser. — As previously stated, the principles involved in 
the flow of water through a diverging tube find a useful appli- 
cation in Mr. Boyden's diffuser. This invention, applied to a 
turbine water wheel 104.25 inches in diameter, and about seven 
hundred horse-power, is represented on Fig. B at X' and on 
Fig. C at X^'. 

The diffuser is supported on iron pillars from below. The 
w'heel is placed suflficiently low to permit the diffuser to be 
submerged at all times when the wheel is in operation, that 
being essential to the most advantageous operation of the 
diffuser. 

When the speed gate is fully raised, the wheel moves w^ith 
the velocity which gives its greatest coetiicient of useful effect. 
On leaving the wheel it necessarily has considerable velocity, 
which would involve a corresponding loss of power, except for 
the effect of the diffuser, which utilizes a portion of it. When 
operating under a fall of thirty-three feet, and the speed gate 
is raised to its full height, this wheel discharges a])out 219 
cubic feet per second. The area of the annular space where 
the water enters the diffuser, is 0.802 x 8.792 <n' = 22.152 square 
feet ; and if the stream passes through this section radially, its 



162 APPENDIX. 

219 
mean velocity must be ~ = 9.886 feet per second, which 
ZZ.xDJi 

is due to a head of 1.519 feet. The area of the annular space 

where the water leaves the difiuser is 1.5 x 15.333 -tt — 72.255 

219 
square feet, and the mean velocity —3.031 feet per 

second, wliich is due to a head of 0.143 feet. According to this 
the saving of head due to the diffnser is 1.519 — 0.143=1.376 

feet, being -—- or about 4|- per cent, of the head available 

ou — i.t) jLy 

without the diifuser, which is equivalent to a gain in the coeffi- 
cient of useful effect to the same extent. Experiments on the 
same turbine, w^ith and without the diffuser, have shown a 
gain due to the latter of about three per cent, in the coefficient 
of useful effect. The diffuser adds to the coefficient of useful 
effect by increasing the velocity of the water passing through 
the wheel, and it must of course increase the quantity of water 
discharged in the same proportion. If it increases the availa- 
ble head three per cent., the velocity, w^hich varies as the 
square root of the head, must be increased in the same propor- 
tion. The power of the wheel, which varies as the product of 
the head into tlie quantity of water discharged, must be 
increased about 4.5 per cent. 



APPENDIX. 163 



explanatio:n" of figures. 

Fig. A. Section through the axis of the Turbine without the Diffuser. 

i Cast-iron Curbs through which the water passes from the wrought-iron 
supply-pipe to the Disc. 

K Cast-iron Disc on which the Guide Curves are fastened. 

L Garniture fitted to lower end of Lower Curb. 

lyi Disc-pipe suspending the Disc from Upper Curb. 

^ Columns of cast-iron supporting Curbs. 

R Regulating Gate of cast-iron, 

S Brackets for raising and lowering the Gate. 

B Wheel. 

C, C Crowns of the Wheel between which the curved buckets are in- 
serted. 

D Main Shaft of the Wheel. 

E' Suspension Box, lined with babbit metal, from which the Wheel hangs 
by the cast-steel Collars around the upper end of the Shaft. 

F Upper portion of Shaft fastened to lower portion, with bearings p' of 
cast-iron lined with babbit metal. 

Q' Gimbal on which the Suspension Box £' rests. 

H Support of the Gimbal. 

O Step to receive cast-steel Pin on lower end of Shaft. 

Fig. A. Plan of Disc, K", Garniture, L', Wheel, C', and Diffuser, M' N'. 

Fig. B, Section of Wheel, Q", Garniture, L "5 Regulating Gate, R", and 
upper and lower Crowns of the Diffuser. 

Fig. D. Diagram for laying out the curves of the Buckets and Guide 
Curves. ' 



164 



APPENDIX. 







APPENDIX. 



165 





